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Proposition 6.84. For every subset \( E \) of a WCG space \( X \) and every \( \bar{x} \in E \) one has\n\n\[{\partial }_{\ell }{d}_{E}\left( \bar{x}\right) \subset {N}_{\ell }\left( {E,\bar{x}}\right)\] | Proof. Let \( {\bar{x}}^{ * } \in {\partial }_{\ell }{d}_{E}\left( \bar{x}\right) \) . By the preceding theorem, there are sequences \( \left( {w}_{n}\right) \rightarrow \bar{x} \) , \( \left( {w}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{x}}^{ * } \) such that \( {w}_{n}^{ * } \in {\partial }_{H}{d}_{E}\left(... | Yes |
Proposition 6.86. If \( g : X \rightarrow Y \) is of class \( {D}^{1} \) at \( \bar{x} \in X \), then \( {D}_{\ell }^{ * }g\left( \bar{x}\right) = {D}_{m}^{ * }g\left( \bar{x}\right) = \) \( {g}^{\prime }{\left( \bar{x}\right) }^{\top } \) . | Proof. Let \( A \mathrel{\text{:=}} {g}^{\prime }\left( \bar{x}\right) \), so that \( {D}_{D}^{ * }g\left( \bar{x}\right) = {A}^{\top } \) . Since \( {D}_{D}^{ * }g\left( \bar{x}\right) \subset {D}_{m}^{ * }g\left( \bar{x}\right) \subset {D}_{\ell }^{ * }g\left( \bar{x}\right) \) , it remains to show that \( {x}^{ * } ... | Yes |
Proposition 6.87. If \( f : X \rightarrow \overline{\mathbb{R}} \) is lower semicontinuous and finite at \( \bar{x} \in X \) and if \( E : X \rightrightarrows \mathbb{R} \) is the multimap with graph \( \operatorname{epi}f \), the limiting directional subdifferential of \( f \) at \( \bar{x} \) satisfies\n\n\[{\partial... | Proof. Given \( {\bar{x}}^{ * } \in {\partial }_{\ell }f\left( \bar{x}\right) \), we can find sequences \( \left( {x}_{n}\right) { \rightarrow }_{f}\bar{x},\left( {x}_{n}^{ * }\right) { \rightarrow }^{ * }{\bar{x}}^{ * } \) such that \( {x}_{n}^{ * } \in {\partial }_{D}f\left( {x}_{n}\right) \) for all \( n \in \mathbb... | Yes |
Proposition 6.88 (Scalarization). Let \( g : X \rightarrow Y \) be continuous at \( \bar{x} \in X \). Then for all \( {y}^{ * } \in {Y}^{ * } \), one has the following inclusion; it is an equality when \( g \) is Lipschitzian near \( \bar{x}, X \) is a WCG Banach space, and \( Y \) is finite-dimensional: | Proof. The proof of the inclusion is similar to that for the subdifferential \( {\partial }_{L} \). Let us prove the opposite inclusion when \( X \) is a WCG space, \( Y \) is finite-dimensional, and \( g \) is Lipschitzian near \( \bar{x} \). Let \( {x}^{ * } \in {D}_{m}^{ * }g\left( \bar{x}\right) \left( {y}^{ * }\ri... | Yes |
Proposition 6.89. (a) If \( f, g \in \mathcal{F}\left( X\right) \) coincide around \( x \), then \( {\partial }_{\ell }f\left( x\right) = {\partial }_{\ell }g\left( x\right) \) . | Proof. (a) is obvious. | No |
Proposition 6.90. Let \( f \mathrel{\text{:=}} h \circ g \), where \( g \mathrel{\text{:=}} \left( {{g}_{1},\ldots ,{g}_{m}}\right) : X \rightarrow {\mathbb{R}}^{m},{g}_{i} \in \mathcal{L}\left( X\right) \) for \( i \in {\mathbb{N}}_{m}, h : {\mathbb{R}}^{m} \rightarrow \mathbb{R} \) is of class \( {C}^{1} \) around \(... | \[ {\partial }_{\ell }f\left( \bar{x}\right) \subset {h}^{\prime }\left( \bar{y}\right) \circ \left( {{\partial }_{\ell }{g}_{1}\left( \bar{x}\right) ,\ldots ,{\partial }_{\ell }{g}_{m}\left( \bar{x}\right) }\right) . \] | Yes |
Proposition 6.91. Let \( X \) and \( Y \) be Banach spaces, let \( f : X \rightarrow \overline{\mathbb{R}} \) be finite and Lipschitzian around \( \bar{x} \in X \), let \( g : Y \rightarrow \mathbb{R} \) be of class \( {D}^{1} \) at \( \bar{y} \in Y \) with \( {g}^{\prime }\left( \bar{y}\right) \neq 0 \) and let \( h \... | \[ \left( {{\bar{x}}^{ * },{\bar{y}}^{ * }}\right) \in \left( {1 - \lambda }\right) {\partial }_{\ell }f\left( \bar{x}\right) \times \lambda {\partial }_{\ell }g\left( \bar{y}\right) \] | Yes |
Proposition 6.92. Let \( V, W \) be Banach spaces, let \( A \in L\left( {V, W}\right) \) with \( W = A\left( V\right) \) , \( \bar{v} \in V,\bar{w} \mathrel{\text{:=}} A\bar{v}, j \in \mathcal{F}\left( V\right), p \in \mathcal{F}\left( W\right) \) such that \( p \circ A \leq j \) . Suppose that for every sequence \( \l... | Proof. Let \( {\bar{w}}^{ * } \in {\partial }_{\ell }p\left( \bar{w}\right) \) . There exist sequences \( \left( {w}_{n}\right) { \rightarrow }_{p}\bar{w},\left( {w}_{n}^{ * }\right) { \rightarrow }^{ * }{\bar{w}}^{ * } \) such that \( {w}_{n}^{ * } \in {\partial }_{D}p\left( {w}_{n}\right) \) for all \( n \in \mathbb{... | Yes |
Proposition 6.93. (a) If \( A \) and \( B \) are closed subsets of Banach spaces \( X \) and \( Y \) respectively, and \( \left( {\bar{x},\bar{y}}\right) \in A \times B \), then \( {N}_{\ell }\left( {A \times B,\left( {\bar{x},\bar{y}}\right) }\right) \subset {N}_{\ell }\left( {A,\bar{x}}\right) \times {N}_{\ell }\left... | Proof. (a) The assertion follows from Proposition 6.89 (e) with \( g \mathrel{\text{:=}} {\iota }_{A}, h \mathrel{\text{:=}} {\iota }_{B} \) . | Yes |
Theorem 6.94 (Sum rule for limiting directional subdifferentials). Let \( X \) be a WCG space, and let \( f = {f}_{1} + \cdots + {f}_{k} \), where \( {f}_{1},\ldots ,{f}_{k} \in \mathcal{L}\left( X\right) \) . Then \[ {\partial }_{\ell }f\left( \bar{x}\right) \subset {\partial }_{\ell }{f}_{1}\left( \bar{x}\right) + \c... | Proof. We know that \( X \) is H-smooth. Let \( {\bar{x}}^{ * } \in {\partial }_{\ell }f\left( \bar{x}\right) \), and let \( \left( {x}_{n}\right) \rightarrow \bar{x},\left( {x}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{x}}^{ * } \) with \( {x}_{n}^{ * } \in {\partial }_{H}f\left( {x}_{n}\right) \) for all \( ... | Yes |
Theorem 6.95 (Chain rule for limiting directional subdifferentials). Let \( X \) and \( Y \) be WCG spaces, let \( g : X \rightarrow Y \) be a map that is Lipschitzian around \( \bar{x} \in X \), and let \( h : Y \rightarrow {\mathbb{R}}_{\infty } \) be Lipschitzian around \( \bar{y} \mathrel{\text{:=}} g\left( \bar{x}... | Proof. Let \( f \mathrel{\text{:=}} h \circ g \) and let \( {\bar{x}}^{ * } \in {\partial }_{\ell }f\left( \bar{x}\right) \) . Let \( r \) (resp. \( s \) ) be the Lipschitz rate of \( f \) (resp. \( h \) ) on a neighborhood of \( \bar{x} \) (resp. \( \bar{y} \) ) and let \( G \) be the graph of \( g \) . Then, by the p... | Yes |
Lemma 7.1. Let \( A : X \rightarrow Y \) be a surjective linear continuous map between two Banach spaces. Then for every \( Z \in \mathcal{S}\left( Y\right) \) there exists some \( W \in \mathcal{S}\left( X\right) \) such that \( A\left( W\right) = Z \) . | Proof. By Michael's selection theorem (Theorem 1.40) there exists a continuous right inverse \( B : Y \rightarrow X \) of \( A \) . Given \( Z \in \mathcal{S}\left( Y\right) \), let \( D \) be a countable dense subset of \( Z \) and let \( W \) be the closure of the set \( E \) of rational combinations of elements of \... | Yes |
Lemma 7.2. Let \( S \) be a closed subset of a Banach space \( X \) and let \( {W}_{0} \in \mathcal{S}\left( X\right) \) . Then there exists \( W \in \mathcal{S}\left( X\right) \) containing \( {W}_{0} \) such that \( d\left( {x, S}\right) = d\left( {x, S \cap W}\right) \) for all \( x \in W \) . | Proof. Starting with \( {W}_{0} \), we define inductively an increasing sequence \( {\left( {W}_{n}\right) }_{n \geq 1} \) of \( \mathcal{S}\left( X\right) \) such that \( d\left( {\cdot, S}\right) = d\left( {\cdot, S \cap {W}_{n}}\right) \) on \( {W}_{n - 1} \) . Assuming that \( {W}_{1},\ldots ,{W}_{n} \) satisfying ... | Yes |
Lemma 7.6. Let \( I \) be a directed set and let \( {\left( {W}_{i}\right) }_{i \in I} \) be a cofinal subfamily of \( \mathcal{S}\left( X\right) \) such that \( {W}_{i} \subset {W}_{j} \) for \( i \leq j \) in I. Given a function \( f : X \rightarrow {\mathbb{R}}_{\infty } \) that is Lipschitzian with rate \( r \) aro... | Proof. We have \( {C}_{j} \subset {C}_{i} \subset r{B}_{{X}^{ * }} \) for \( i, j \in I \) satisfying \( i \leq j \) . Let \( {\bar{x}}^{ * } \) be a weak* cluster point of \( {\left( {x}_{j}^{ * }\right) }_{j \in I} \) in the weak* compact set \( r{B}_{{X}^{ * }} \) . Since for all \( i \in I \) the set \( {C}_{i} \) ... | Yes |
Proposition 7.8. For a locally Lipschitzian function \( f \) on an Asplund space \( X \) one has\n\n\[{\partial }_{L}f\left( \bar{x}\right) \subset {\partial }_{\ell }f\left( \bar{x}\right) \subset {\partial }_{G}f\left( \bar{x}\right) \subset {\overline{\operatorname{co}}}^{ * }\left( {{\partial }_{G}f\left( \bar{x}\r... | Proof. The inclusion \( {\partial }_{L}f\left( \bar{x}\right) \subset {\partial }_{\ell }f\left( \bar{x}\right) \) (resp. \( {\partial }_{\ell }f\left( \bar{x}\right) \subset {\partial }_{G}f\left( \bar{x}\right) \) ) stems from the obvious inclusion \( {\partial }_{F}f\left( x\right) \subset {\partial }_{D}f\left( x\r... | Yes |
Lemma 7.10. Let \( X \) be a separable Banach space, let \( \left( {X}_{n}\right) \) be an increasing sequence of linear subspaces whose union is dense in \( X \), and let \( f \in \mathcal{L}\left( X\right) ,\bar{x} \in X \) , \( {\bar{x}}^{ * } \in {X}^{ * } \) . Let \( \left( {x}_{n}\right) \rightarrow \bar{x} \) in... | Proof. We first observe that we may assume that for all \( n \in \mathbb{N},{X}_{n} \) is finite-dimensional. If it is not the case, taking a countable subset \( \left\{ {{e}_{k} : k \in \mathbb{N}}\right\} \) and for \( n \in \mathbb{N} \) some \( p\left( n\right) \in \mathbb{N} \) and \( {x}_{k, n} \in {X}_{p\left( n... | Yes |
Proposition 7.12. Let \( X, Y \) be two Banach spaces, let \( A : X \rightarrow Y \) be a surjective linear continuous map, let \( f \in \mathcal{L}\left( X\right), g \in \mathcal{L}\left( Y\right) \) be such that \( f \geq g \circ A \) . Suppose that for some \( \bar{x} \in X \) and every sequence \( \left( {y}_{n}\ri... | Proof. Let \( {\bar{y}}^{ * } \in {\partial }_{G}g\left( \bar{y}\right) \), let \( {\bar{x}}^{ * } \mathrel{\text{:=}} {A}^{\top }\left( {\bar{y}}^{ * }\right) \), and let \( W \in \mathcal{S}\left( X\right) \) . One has \( Z \mathrel{\text{:=}} \operatorname{cl}A\left( W\right) \in \) \( \mathcal{S}\left( Y\right) \),... | Yes |
Lemma 7.13. Let \( E \) be a nonempty subset of a Banach space \( X \), let \( g \mathrel{\text{:=}} {d}_{E},\bar{x} \in X \) , \( {\bar{x}}^{ * } \in {\partial }_{G}g\left( \bar{x}\right) \), and let \( {W}_{0} \in \mathcal{S}\left( X\right) \) . Then there exist some \( W \in \mathcal{S}\left( X\right) \) containing ... | Proof. We construct inductively a sequence \( \left( {W}_{n}\right) \) in \( \mathcal{S}\left( X\right) \) such that \( {W}_{n} \subset {W}_{n + 1} \) , \( d\left( {\cdot, E}\right) = d\left( {\cdot, E \cap {W}_{n + 1}}\right) \) on \( {W}_{n + 1},\bar{x} \in {W}_{n + 1} \) for all \( n \) . For \( n = 0,{W}_{1} \) is ... | Yes |
Proposition 7.14. Let \( X, Y \) be two Banach spaces, let \( A : X \rightarrow Y \) be a surjective linear continuous map, let \( E \) be a closed subset of \( Y \), and let \( f \in \mathcal{L}\left( X\right) \) be such that \( f \geq g \circ A \) for \( g \mathrel{\text{:=}} {d}_{E} \) . Let \( \bar{x} \in {A}^{-1}\... | Proof. Let \( {\bar{y}}^{ * } \in {\partial }_{G}g\left( \bar{y}\right) \), let \( {\bar{x}}^{ * } \mathrel{\text{:=}} {A}^{\top }\left( {\bar{y}}^{ * }\right) \), and let \( W \in \mathcal{S}\left( X\right) \) containing \( \bar{x} \) . Let us show that \( {r}_{W}\left( {\bar{x}}^{ * }\right) \in {\partial }_{\ell }^{... | Yes |
Proposition 7.15. (a) If \( f, g \in \mathcal{L}\left( X\right) \) coincide around \( x \), then \( {\partial }_{G}f\left( x\right) = {\partial }_{G}g\left( x\right) \) . | Proof. (a) is obvious. | No |
Proposition 7.17. Let \( X, Y \) be Banach spaces, let \( f : X \rightarrow \overline{\mathbb{R}} \) be finite and Lipschitz-ian around \( \bar{x} \), let \( g : Y \rightarrow \mathbb{R} \) be of class \( {D}^{1} \) around \( \bar{y} \) with \( {g}^{\prime }\left( \bar{y}\right) \neq 0 \), let \( \left( {\bar{x},\bar{y... | Proof. Let \( \left( {{\bar{x}}^{ * },0}\right) \in {\partial }_{G}h\left( {\bar{x},\bar{y}}\right) \) . For all \( W \in \mathcal{S}\left( X\right), Z \in \mathcal{S}\left( Y\right) \) we have \( \left( {{\bar{x}}^{ * },0}\right) \in \) \( {\partial }_{\ell, X \times Y}^{W \times Z}h\left( {\bar{x},\bar{y}}\right) \) ... | Yes |
Proposition 7.18. Let \( S \) be a closed subset \( S \) of a Banach space \( X \) and let \( \bar{x} \in S \) .\n\n(a) The normal cone \( {N}_{G}\left( {S,\bar{x}}\right) \) to \( S \) at \( \bar{x} \) does not depend on the choice of the norm on \( X \) among those inducing the topology of \( X \) . | Proof. (a) The result follows from Proposition 7.14, taking for \( A \) the identity map and \( f = c{d}_{S}^{\prime } \) where \( c > 0 \) and \( {d}_{S}^{\prime } \) is the distance associated with an equivalent norm \( \parallel \cdot {\parallel }^{\prime } \) . | Yes |
Proposition 7.19. Let \( X, Y \) be Banach spaces, let \( \bar{x} \in E \subset X, F \subset Y \) and let \( A \in \) \( L\left( {X, Y}\right) \) be such that \( A\left( E\right) \subset F \) . Suppose that for every sequence \( \left( {y}_{n}\right) \rightarrow \bar{y} \mathrel{\text{:=}} A\bar{x} \) in \( F \) there ... | Proof. Setting \( f \mathrel{\text{:=}} \parallel A\parallel {d}_{E}, g \mathrel{\text{:=}} {d}_{F} \), let us observe that for all \( x \in X \), we have\n\n\[ \ng\left( {Ax}\right) \leq \mathop{\inf }\limits_{{u \in E}}\parallel {Ax} - {Au}\parallel \leq \mathop{\inf }\limits_{{u \in E}}\parallel A\parallel \parallel... | Yes |
Proposition 7.20. For every locally Lipschitzian function \( f \) on a Banach space \( X \) and \( \bar{x} \in X,\bar{r} \mathrel{\text{:=}} f\left( \bar{x}\right) ,\bar{e} \mathrel{\text{:=}} \left( {\bar{x}, f\left( \bar{x}\right) }\right) \), one has\n\n\[ \n{\partial }_{G}f\left( \bar{x}\right) = \left\{ {{x}^{ * }... | Proof. Let \( f \in \mathcal{L}\left( X\right) ,\bar{x} \in X \) and let \( E \) be the epigraph of \( f \) . Without loss of generality we may assume that \( f \) is globally Lipschitzian with rate \( c > 0 \) and even that \( c = 1 \) (since we can change the norm of \( X \) to the norm \( c\parallel \cdot \parallel ... | Yes |
Proposition 7.22. For every closed subset \( S \) of \( X \) and for every \( \bar{x} \in S \) one has\n\n\[{N}_{G}\left( {S,\bar{x}}\right) = {\partial }_{G}{\iota }_{S}\left( \bar{x}\right)\] | Proof. The epigraph \( E \) of \( {\iota }_{S} \) is just \( S \times \mathbb{R} \), so that taking the sum norm on \( X \times \mathbb{R} \) , one has \( {d}_{E}\left( {x, r}\right) = {d}_{S}\left( x\right) + {r}^{ - } \), where \( {r}^{ - } \mathrel{\text{:=}} \max \left( {-r,0}\right) \) . Since \( {d}_{S} \) and \(... | Yes |
Corollary 7.24. Let \( g, h \in \mathcal{F}\left( X\right) \) , \( h \) being circa-differentiable at \( \bar{x} \in \operatorname{dom}g \cap \operatorname{dom}h \) . Then\n\n\[{\partial }_{G}\left( {g + h}\right) \left( \bar{x}\right) = {\partial }_{G}g\left( \bar{x}\right) + {h}^{\prime }\left( \bar{x}\right) .\] | Proof. Since \( h \) is Lipschitzian around \( \bar{x} \) with \( {\partial }_{G}h\left( \bar{x}\right) = \left\{ {{h}^{\prime }\left( \bar{x}\right) }\right\} \) and \( g = f - h \) for \( f \mathrel{\text{:=}} g + h \), one has \( {\partial }_{G}f\left( \bar{x}\right) \subset {\partial }_{G}g\left( \bar{x}\right) + {... | Yes |
Proposition 7.26. For every Banach space \( X \), for every closed subset \( S \) of \( X \), every \( \bar{x} \in S \), and every lower semicontinuous function \( f \) on \( X \) one has\n\n\[ \n{N}_{C}\left( {S,\bar{x}}\right) = {\overline{\operatorname{co}}}^{ * }\left( {{N}_{G}\left( {S,\bar{x}}\right) }\right) ,\;... | Proof. Let \( {\bar{x}}^{ * } \in {N}_{G}\left( {S,\bar{x}}\right) \), so that there exists some \( r \in \lbrack 1, + \infty ) \) such that \( {r}^{-1}{\bar{x}}^{ * } \in \) \( {\partial }_{G}{d}_{S}\left( \bar{x}\right) \subset {\partial }_{C}{d}_{S}\left( \bar{x}\right) \), whence \( {\bar{x}}^{ * } \in {\mathbb{R}}... | Yes |
Is Light a Wave or a Particle? | Beginning in the late seventeenth century and continuing into the early eighteenth century, there was a vigorous debate in the scientific community over the nature of light. One camp, following the views of Isaac Newton, claimed that light consisted of a group of particles or \ | No |
Example 2.2 (Harmonic Oscillator) If the force is given by Hooke's law, \( F\left( x\right) = - {kx} \), where \( k \) is a positive constant, then Newton’s law can be written as \( m\ddot{x} + {kx} = 0 \). | The general solution of this equation is\n\n\[ \n x\left( t\right) = a\cos \left( {\omega t}\right) + b\sin \left( {\omega t}\right) \n\]\n\nwhere \( \omega \mathrel{\text{:=}} \sqrt{k/m} \) is the frequency of oscillation. | Yes |
Theorem 2.3 Suppose a particle satisfies Newton’s law in the form \( m\ddot{x} = \) \( F\left( x\right) \) . Let \( V \) and \( E \) be as in (2.2) and (2.3). Then the energy \( E \) is conserved, meaning that for each solution \( x\left( t\right) \) of Newton’s law, \( E\left( {x\left( t\right) ,\dot{x}\left( t\right)... | Proof. We verify this by differentiation, using the chain rule:\n\n\[ \frac{d}{dt}E\left( {x\left( t\right) ,\dot{x}\left( t\right) }\right) = \frac{d}{dt}\left( {\frac{1}{2}m{\left( \dot{x}\left( t\right) \right) }^{2} + V\left( {x\left( t\right) }\right) }\right) \]\n\n\[ = m\dot{x}\left( t\right) \ddot{x}\left( t\ri... | Yes |
Proposition 2.4 Suppose a particle moves in the presence of a force law given by \( F\left( {x,\dot{x}}\right) = {F}_{1}\left( x\right) - \gamma \dot{x} \), with \( \gamma > 0 \) . Define the energy \( E \) of the system by\n\n\[ E\left( {x,\dot{x}}\right) = \frac{1}{2}m{\dot{x}}^{2} + V\left( x\right) \]\n\nwhere \( {... | Proof. We differentiate as in the proof of Theorem 2.3, except that now \( {dV}/{dx} = - {F}_{1}\left( x\right) \) :\n\n\[ \frac{d}{dt}E\left( {x\left( t\right) ,\dot{x}\left( t\right) }\right) = \dot{x}\left( t\right) \left\lbrack {m\ddot{x}\left( t\right) - {F}_{1}\left( {x\left( t\right) }\right) }\right\rbrack . \]... | Yes |
Consider Newton's law (2.7) in the case of a velocity-independent force: \( m\ddot{\mathbf{x}}\left( t\right) = \mathbf{F}\left( {\mathbf{x}\left( t\right) }\right) \) . Then an energy function of the form\n\n\[ E\left( {\mathbf{x},\dot{\mathbf{x}}}\right) = \frac{1}{2}m{\left| \dot{\mathbf{x}}\right| }^{2} + V\left( \... | Proof. Differentiating gives\n\n\[ \frac{d}{dt}\left( {\frac{1}{2}m{\left| \dot{\mathbf{x}}\left( t\right) \right| }^{2} + V\left( {\mathbf{x}\left( t\right) }\right) }\right) = m\mathop{\sum }\limits_{{j = i}}^{n}{\dot{x}}_{j}\left( t\right) {\ddot{x}}_{j}\left( t\right) + \mathop{\sum }\limits_{{j = 1}}^{n}\frac{\par... | Yes |
Proposition 2.7 Suppose \( U \) is a simply connected domain in \( {\mathbb{R}}^{n} \) and \( \mathbf{F} \) is a smooth, \( {\mathbb{R}}^{n} \) -valued function on \( U \) . Then \( \mathbf{F} \) is conservative if and only if \( \mathbf{F} \) satisfies\n\n\[ \frac{\partial {F}_{j}}{\partial {x}_{k}} - \frac{\partial {... | Proof. If \( \mathbf{F} \) is conservative, then\n\n\[ \frac{\partial {F}_{j}}{\partial {x}_{k}} = - \frac{{\partial }^{2}V}{\partial {x}_{k}\partial {x}_{j}} = - \frac{{\partial }^{2}V}{\partial {x}_{j}\partial {x}_{k}} = \frac{\partial {F}_{k}}{\partial {x}_{j}} \]\n\nat every point in \( U \) . In the other directio... | Yes |
Proposition 2.8 Suppose a particle in \( {\mathbb{R}}^{n} \) moves in the presence of a force \( \mathbf{F} \) of the form\n\n\[ \mathbf{F}\left( {\mathbf{x},\mathbf{v}}\right) = - \nabla V\left( \mathbf{x}\right) + {\mathbf{F}}_{2}\left( {\mathbf{x},\mathbf{v}}\right) \]\n\nwhere \( V \) is a smooth function and where... | Proof. See Exercise 8. | No |
Proposition 2.9 An energy function of the form (2.10) is constant along each trajectory if\n\n\[ \n{\nabla }^{j}V = - {\mathbf{F}}^{j} \n\]\n\n(2.11)\n\nfor each \( j \), where \( {\nabla }^{j} \) is the gradient with respect to the variable \( {\mathbf{x}}^{j} \) . | Proof. We compute that\n\n\[ \n\frac{dE}{dt} = \mathop{\sum }\limits_{{j = 1}}^{N}\left\lbrack {{m}_{j}{\dot{\mathbf{x}}}^{j} \cdot {\ddot{\mathbf{x}}}^{j} + {\nabla }^{j}V \cdot {\dot{\mathbf{x}}}^{j}}\right\rbrack \n\]\n\n\[ \n= \mathop{\sum }\limits_{{j = 1}}^{N}{\dot{\mathbf{x}}}^{j} \cdot \left\lbrack {{m}_{j}{\dd... | Yes |
Proposition 2.10 Suppose a force function \( \mathbf{F} = \left( {{\mathbf{F}}^{1},\ldots ,{\mathbf{F}}^{N}}\right) \) is defined on a simply connected domain \( U \) in \( {\mathbb{R}}^{nN} \) . Then there exists a smooth function \( V \) on \( U \) satisfying\n\n\[ \n{\nabla }^{j}V = - {\mathbf{F}}^{j} \n\]\n\nfor al... | Proof. Apply Proposition 2.7 with \( n \) replaced by \( {nN} \) and with \( j \) and \( k \) replaced by the pairs \( \left( {j, k}\right) \) and \( \left( {l, m}\right) \) . | Yes |
Proposition 2.13 If a multiparticle system has a force law coming from a potential \( V \), then the total momentum of the system is conserved if and only if\n\n\[ V\left( {{\mathbf{x}}^{1} + \mathbf{a},{\mathbf{x}}^{2} + \mathbf{a},\ldots ,{\mathbf{x}}^{N} + \mathbf{a}}\right) = V\left( {{\mathbf{x}}^{1},{\mathbf{x}}^... | Proof. Apply (2.14) with \( \mathbf{a} = t{\mathbf{e}}_{k} \), where \( {\mathbf{e}}_{k} \) is the vector with a 1 in the \( k \) th spot and zeros elsewhere. Differentiating with respect to \( t \) at \( t = 0 \) gives\n\n\[ 0 = \mathop{\sum }\limits_{{j = 1}}^{N}\frac{\partial V}{\partial {x}_{k}^{j}} = - \mathop{\su... | Yes |
Proposition 2.15 Suppose the total momentum \( \mathbf{p} \) of a system is conserved. Then the center of mass moves in a straight line at constant speed. Specifically,\n\n\[ \mathbf{c}\left( t\right) = \mathbf{c}\left( {t}_{0}\right) + \left( {t - {t}_{0}}\right) \frac{\mathbf{p}}{M} \]\n\nwhere \( \mathbf{c}\left( {t... | Proof. The result follows easily from (2.15). ∎ | No |
Proposition 2.18 Suppose a particle of mass \( m \) is moving in \( {\mathbb{R}}^{2} \) under the influence of a conservative force with the potential function \( V\left( \mathbf{x}\right) \) . If \( V \) is invariant under rotations in \( {\mathbb{R}}^{2} \), then the angular momentum \( J = \) \( {x}_{1}{p}_{2} - {x}... | Proof. Differentiating (2.16) along a solution of Newton's law gives\n\n\[ \frac{dJ}{dt} = \frac{d{x}_{1}}{dt}{p}_{2} + {x}_{1}\frac{d{p}_{2}}{dt} - \frac{d{x}_{2}}{dt}{p}_{1} - {x}_{2}\frac{d{p}_{1}}{dt} \]\n\n\[ = \frac{1}{m}{p}_{1}{p}_{2} - {x}_{1}\frac{\partial V}{\partial {x}_{2}} - \frac{1}{m}{p}_{2}{p}_{1} + {x}... | Yes |
Proposition 2.23 For all smooth functions \( f, g \), and \( h \) on \( {\mathbb{R}}^{2n} \) we have the following:\n\n1. \( \{ f, g + {ch}\} = \{ f, g\} + c\{ f, h\} \) for all \( c \in \mathbb{R} \)\n\n2. \( \{ g, f\} = - \{ f, g\} \)\n\n3. \( \{ f,{gh}\} = \{ f, g\} h + g\{ f, h\} \)\n\n4. \( \{ f,\{ g, h\} \} = \{ ... | Proof. The first two properties of the Poisson bracket are obvious and the third is an easy consequence of the product rule. Let us think about what goes into proving Property 4 by direct computation. (An alternative proof is given in Exercise 15.) We compute that\n\n\[ \{ f,\{ g, h\} \} = \mathop{\sum }\limits_{{j = 1... | No |
Proposition 2.24 The position and momentum functions satisfy the following Poisson bracket relations:\n\n\[ \left\{ {{x}_{j},{x}_{k}}\right\} = 0 \]\n\n\[ \left\{ {{p}_{j},{p}_{k}}\right\} = 0 \]\n\n\[ \left\{ {{x}_{j},{p}_{k}}\right\} = {\delta }_{jk} \] | Proof. Direct calculation. - | Yes |
Proposition 2.25 If \( \left( {\mathbf{x}\left( t\right) ,\mathbf{p}\left( t\right) }\right) \) is a solution to Hamilton’s equation (2.25), then for any smooth function \( f \) on \( {\mathbb{R}}^{2n} \) we have\n\n\[ \frac{d}{dt}f\left( {\mathbf{x}\left( t\right) ,\mathbf{p}\left( t\right) }\right) = \{ f, H\} \left(... | Proof. Using the chain rule and Hamilton's equations, we have\n\n\[ \frac{df}{dt} = \mathop{\sum }\limits_{{j = 1}}^{n}\left( {\frac{\partial f}{\partial {x}_{j}}\frac{d{x}_{j}}{dt} + \frac{\partial f}{\partial {p}_{j}}\frac{d{p}_{j}}{dt}}\right) \]\n\n\[ = \mathop{\sum }\limits_{{j = 1}}^{n}\left( {\frac{\partial f}{\... | Yes |
Call a smooth function \( f \) on \( {\mathbb{R}}^{2n} \) a conserved quantity if \( f\left( {\mathbf{x}\left( t\right) ,\mathbf{p}\left( t\right) }\right) \) is independent of \( t \) for each solution \( \left( {\mathbf{x}\left( t\right) ,\mathbf{p}\left( t\right) }\right) \) of Hamilton’s equations. Then \( f \) is ... | \[ \{ f, H\} = 0. \] | Yes |
Theorem 2.27 (Liouville's Theorem) The flow associated with Hamilton’s equations, for an arbitrary Hamiltonian function \( H \), preserves the (2n)-dimensional volume measure\n\n\[ d{x}_{1}d{x}_{2}\cdots d{x}_{n}d{p}_{1}d{p}_{2}\cdots d{p}_{n} \]\n\nWhat this means, more precisely, is that if a measurable set \( E \) i... | Proof. Hamilton's equations may be written as\n\n\[ \frac{d}{dt}\left\lbrack \begin{matrix} {x}_{1} \\ \vdots \\ {x}_{n} \\ {p}_{1} \\ \vdots \\ {p}_{n} \end{matrix}\right\rbrack = \left\lbrack \begin{matrix} \frac{\partial H}{\partial {p}_{1}} \\ \vdots \\ \frac{\partial H}{\partial {p}_{n}} \\ - \frac{\partial H}{\pa... | Yes |
The Hamiltonian flow generated by the function\n\n\[ \n{f}_{\mathbf{a}}\left( {\mathbf{x},\mathbf{p}}\right) \mathrel{\text{:=}} \mathbf{a} \cdot \mathbf{p} \n\]\n\n(2.30) | is given by\n\n\[ \n\mathbf{x}\left( t\right) = {\mathbf{x}}_{0} + t\mathbf{a} \n\]\n\n\[ \n\mathbf{p}\left( t\right) = {\mathbf{p}}_{0} \n\]\n\n(2.31)\n\nand the Hamiltonian flow generated by the function\n\n\[ \n{g}_{\mathbf{b}}\left( {\mathbf{x},\mathbf{p}}\right) \mathrel{\text{:=}} \mathbf{b} \cdot \mathbf{x} \n\]... | Yes |
For a particle moving in \( {\mathbb{R}}^{2} \), the Hamiltonian flow generated by the angular momentum function\n\n\[ J\left( {\mathbf{x},\mathbf{p}}\right) = {x}_{1}{p}_{2} - {x}_{2}{p}_{1} \]\n\nconsists of simultaneous rotations of \( \mathbf{x} \) and \( \mathbf{p} \) . That is to say,\n\n\[ \left\lbrack \begin{ar... | Proof. If we plug the angular momentum function \( J \) into Hamilton’s equations in place of \( H \), we obtain\n\n\[ \frac{d{x}_{1}}{dt} = \frac{\partial J}{\partial {p}_{1}} = - {x}_{2};\;\frac{d{p}_{1}}{dt} = - \frac{\partial J}{\partial {x}_{1}} = - {p}_{2} \]\n\n\[ \frac{d{x}_{2}}{dt} = \frac{\partial J}{\partial... | Yes |
Proposition 2.32 Consider the phase space for a system of \( N \) particles moving in \( {\mathbb{R}}^{n} \), namely \( {\mathbb{R}}^{2nN} \), thought of as the set of \( \left( {2N}\right) \) -tuples of the form\n\n\[ \left( {{\mathbf{x}}^{1},\ldots ,{\mathbf{x}}^{N},{\mathbf{p}}^{1},\ldots ,{\mathbf{p}}^{N}}\right) \... | The proof of these results is entirely similar to the one-particle case and is omitted. | No |
Proposition 2.34 The Runge-Lenz vector is conserved quantity for Newton's law with force given by (2.35). | Proof. Since \( \mathbf{J} \) is conserved, we compute that\n\n\[ \dot{\mathbf{A}}\left( t\right) = \frac{1}{mk}\mathbf{F} \times \mathbf{J} - \frac{1}{\left| \mathbf{x}\right| }\frac{\mathbf{p}}{m} + \frac{\mathbf{x}}{{\left| \mathbf{x}\right| }^{2}}\mathop{\sum }\limits_{{j = 1}}^{3}\frac{\partial \left| \mathbf{x}\r... | Yes |
Proposition 2.35 The magnitude of the Runge-Lenz vector \( \mathbf{A} \) satisfies\n\n\[ \n{\left| \mathbf{A}\right| }^{2} = 1 + \frac{2{\left| \mathbf{J}\right| }^{2}}{m{k}^{2}}E \n\]\n\nwhere \( E = {\left| \mathbf{p}\right| }^{2}/\left( {2m}\right) - k/\left| \mathbf{x}\right| \) is the energy of the particle. Furth... | Proof. Using the identity \( \mathbf{b} \cdot \left( {\mathbf{c} \times \mathbf{d}}\right) = \mathbf{d} \cdot \left( {\mathbf{b} \times \mathbf{c}}\right) \), we see that\n\n\[ \n\widehat{\mathbf{x}} \cdot \left( {\mathbf{p} \times \mathbf{J}}\right) = \mathbf{J} \cdot \left( {\widehat{\mathbf{x}} \times \mathbf{p}}\ri... | Yes |
Proposition 2.37 If \( A \mathrel{\text{:=}} \left| \mathbf{A}\right| < 1 \) ,(2.38) is the equation of an ellipse with eccentricity \( A \) and with the origin being one focus of the ellipse. If \( A > 1 \) , (2.38) is the equation of a hyperbola, and if \( A = 1 \) ,(2.38) is the equation of a parabola. | Proof. We continue to work in a coordinate system in which \( \mathbf{A} \) is along the positive \( {x}_{1} \) -axis. Then (2.38) becomes\n\n\[ \sqrt{{x}^{2} + {y}^{2}} = \alpha \frac{1}{1 + A\frac{x}{\sqrt{{x}^{2} + {y}^{2}}}} \]\n\nwhere \( \alpha = {\left| \mathbf{J}\right| }^{2}/\left( {mk}\right) \) . From this w... | Yes |
Proposition 3.4 Suppose \( A \) is a symmetric operator on \( \mathbf{H} \). 1. For all \( \psi \in \operatorname{Dom}\left( A\right) \), the quantity \( \langle \psi ,{A\psi }\rangle \) is real. More generally, if \( \psi ,{A\psi },\ldots ,{A}^{m - 1}\psi \) all belong to \( \operatorname{Dom}\left( A\right) \), then ... | Proof. Since \( A \) is symmetric, we have \[ \langle \psi ,{A\psi }\rangle = \langle {A\psi },\psi \rangle = \overline{\langle \psi ,{A\psi }\rangle } \] for all \( \psi \in \operatorname{Dom}\left( A\right) \). If \( \psi ,{A\psi },\ldots ,{A}^{m - 1}\psi \) all belong to the domain of \( A \) , we can use the symmet... | Yes |
Proposition 3.5 (de Broglie hypothesis) If the wave function of a particle has spatial frequency \( k \), then the momentum \( p \) of the particle is\n\n\[ p = \hslash k \]\n\nwhere \( \hslash \) is Planck’s constant. | The Davisson-Germer electron-diffraction experiments, described in Sect. 1.2.3, strongly support not only the idea that electrons have wavelike behavior, but also the specific relationship (3.4) between the momentum of an electron and the spatial frequency of the associated wave. Of course, Proposition 3.5 is rather va... | No |
Define the momentum operator \( P \) by\n\n\[ P = - i\hslash \frac{d}{dx}. \]\n\nThen for all sufficiently nice unit vectors \( \psi \) in \( {L}^{2}\left( \mathbb{R}\right) \), we have\n\n\[ \left\langle {\psi ,{P}^{m}\psi }\right\rangle = {\int }_{-\infty }^{\infty }{\left( \hslash k\right) }^{m}{\left| \widehat{\psi... | Proof. If \( \psi \) is in, say, the Schwartz space (Definition A.15), then, by applying Proposition A. \( {17}\mathrm{\;m} \) times, we see that the Fourier transform of the \( n \) th derivative of \( \psi \) is \( {\left( ik\right) }^{m}\widehat{\psi }\left( k\right) \), and so the Fourier transform of \( {P}^{m}\ps... | Yes |
Proposition 3.8 The position and momentum operators \( X \) and \( P \) do not commute, but satisfy the relation\n\n\[ {XP} - {PX} = i\hslash I \] | Proof. Using the product rule we calculate that\n\n\[ {PX\psi } = - i\hslash \frac{d}{dx}\left( {{x\psi }\left( x\right) }\right) \]\n\n\[ = - i\hslash \psi \left( x\right) - i\hslash x\frac{d\psi }{dx} \]\n\n\[ = - i\hslash \psi \left( x\right) + {XP\psi } \]\n\nfrom which (3.14) follows. | Yes |
Proposition 3.9 For all sufficiently nice functions \( \phi \) and \( \psi \) in \( {L}^{2}\left( \mathbb{R}\right) \) , we have\n\n\[ \langle \phi ,{X\psi }\rangle = \langle {X\phi },\psi \rangle \]\n\nand\n\n\[ \langle \phi ,{P\psi }\rangle = \langle {P\phi },\psi \rangle \] | Proof. Suppose that \( \phi \) and \( \psi \) belong to \( {L}^{2}\left( \mathbb{R}\right) \) and that the functions \( {x\phi }\left( x\right) \) and \( {x\psi }\left( x\right) \) also belong to \( {L}^{2}\left( \mathbb{R}\right) \) . Then since \( x \) is real, we have\n\n\[ {\int }_{-\infty }^{\infty }\overline{\phi... | Yes |
Proposition 3.11 (Eigenvectors) If a quantum system is in a state described by a unit vector \( \psi \in \mathbf{H} \) and for some quantum observable \( \widehat{f} \) we have \( \widehat{f}\psi = {\lambda \psi } \) for some \( \lambda \in \mathbb{R} \), then\n\n\[ E\left( {f}^{m}\right) = {\left\langle {\left( \wideh... | Proof. The relation (3.18) follows from (3.15) and the fact that \( \widehat{f}\psi = \) \( {\lambda \psi } \) . Meanwhile, if \( \mu \) is the \( \delta \) -measure at \( \lambda \), then certainly (3.19) holds. Meanwhile, since the \( m \) th moment grows only exponentially with \( m \), even the most elementary uniq... | Yes |
Suppose \( \widehat{f} \) has an orthonormal basis \( \left\{ {e}_{j}\right\} \) of eigenvectors with distinct (real) eigenvalues \( {\lambda }_{j} \). Suppose also that \( \psi \) is a unit vector in \( \mathbf{H} \) with the expansion\n\n\[ \psi = \mathop{\sum }\limits_{{j = 1}}^{\infty }{a}_{j}{e}_{j} \]\n\nThen for... | Assuming that \( \psi \) is in the domain of \( {\left( \widehat{f}\right) }^{m} \), it is easy to verify that the probabilities in (3.21) are consistent with the expectation values given in Axiom 3. After all, if \( \psi \) is given as in (3.20), then we can readily calculate that \( \left\langle {\psi ,{\left( \wideh... | No |
Proposition 3.14 Suppose \( \psi \left( t\right) \) is a solution of the Schrödinger equation and \( A \) is a self-adjoint operator on \( \mathbf{H} \) . Assuming certain natural domain conditions hold, we have\n\n\[ \frac{d}{dt}\langle A{\rangle }_{\psi \left( t\right) } = {\left\langle \frac{1}{i\hslash }\left\lbrac... | Proof. Let \( \psi \left( t\right) \) be a solution to the Schrödinger equation and let us compute at first without worrying about domains of the operators involved. If we use the product rule (Exercise 1) for differentiation of the inner product, we obtain\n\n\[ \frac{d}{dt}\langle \psi \left( t\right) ,{A\psi }\left(... | No |
Proposition 3.15 For any vector space \( V \) over \( \mathbb{C} \) and linear operators \( A \) , \( B \), and \( C \) on \( V \), the following relations hold.\n\n\[ \text{1.}\left\lbrack {A, B + {\alpha C}}\right\rbrack = \left\lbrack {A, B}\right\rbrack + \alpha \left\lbrack {A, C}\right\rbrack \text{for all}\alpha... | Proof. The first two properties of the commutator are obvious, and the third is easily verified by writing things out. Property 4 can also be proved by writing things out, but it is slightly messier. Each of the three double commutators on the left-hand side of (3.30) generates four terms, for a total of 12 terms. Each... | No |
Proposition 3.16 If \( \phi \left( t\right) \) and \( \psi \left( t\right) \) are solutions to the Schrödinger equation (3.28), the quantity \( \langle \phi \left( t\right) ,\psi \left( t\right) \rangle \) is independent of \( t \) . In particular, \( \parallel \psi \left( t\right) \parallel \) is independent of \( t \... | Proof. Using again the product rule, we have\n\n\[ \frac{d}{dt}\langle \phi \left( t\right) ,\psi \left( t\right) \rangle = \left\langle {\frac{1}{i\hslash }\widehat{H}\phi \left( t\right) ,\psi \left( t\right) }\right\rangle + \left\langle {\phi \left( t\right) ,\frac{1}{i\hslash }\widehat{H}\psi \left( t\right) }\rig... | Yes |
Lemma 3.22 Suppose \( A \) is a self-adjoint operator on \( \mathbf{H} \) and that \( A\left( \cdot \right) \) is a solution to (3.39) with \( A\left( 0\right) = A \) . Then for any positive integer \( m \), the map\n\n\[ \n t \mapsto {\left( A\left( t\right) \right) }^{m} \n\]\n\nis also a solution to (3.39). | Proof. If we use (3.40), then the result holds because\n\n\[ \n{e}^{{it}\widehat{H}/\hslash }{A}^{m}{e}^{-{it}\widehat{H}/\hslash } = {e}^{{it}\widehat{H}/\hslash }A{e}^{-{it}\widehat{H}/\hslash }{e}^{{it}\widehat{H}/\hslash }A{e}^{-{it}\widehat{H}/\hslash }\cdots {e}^{{it}\widehat{H}/\hslash }A{e}^{-{it}\widehat{H}/\h... | Yes |
Proposition 3.23 Suppose the Hamiltonian of a quantum system is as in Proposition 3.21. Then the operators \( X\\left( t\\right) \) and \( P\\left( t\\right) \) defined by (3.39) satisfy the following operator-valued differential equation:\n\n\[ \n\\frac{dX}{dt} = \\frac{1}{m}P\\left( t\\right) \n\]\n\n\[ \n\\frac{dP}{... | Proof. See Exercise 7. ∎ | No |
Proposition 3.24 The following functions are solutions to (3.43) satisfying the boundary conditions (3.44):\n\n\[ \n{\psi }_{j}\left( x\right) = \sqrt{\frac{2}{L}}\sin \left( \frac{j\pi x}{L}\right) ,\;j = 1,2,3,\ldots ,\n\]\n\nand the corresponding eigenvalues \( {E}_{j} \) are given by (3.47). The functions \( {\psi ... | Proof. We have already verified the equation and eigenvalue for each \( {\psi }_{j} \) . It is a simple computation to verify that the \( {\psi }_{j} \) ’s are orthonormal, and the elementary theory of Fourier series (Fourier sine series, in this case) shows that the \( {\psi }_{j} \) ’s form an orthonormal basis for \... | Yes |
Proposition 4.1 The phase velocity of a particle with momentum \( p = \hslash k \) is\n\n\[ \text{phase velocity} = \frac{\omega \left( k\right) }{k} = \frac{\hslash k}{2m} = \frac{p}{2m}\text{.} \] | This velocity is half the velocity of a classical particle of momentum \( p \) . | No |
Proposition 4.3 If \( \psi \left( {x, t}\right) \) is as in Proposition 4.2, then the Fourier transform of \( \psi \left( {x, t}\right) \), with respect to \( x \) with \( t \) fixed, is given by \[ \widehat{\psi }\left( {k, t}\right) = {\widehat{\psi }}_{0}\left( k\right) \exp \left\lbrack {-i\frac{\hslash {k}^{2}t}{2... | Proof. We can write (4.5) as \[ \psi \left( {x, t}\right) = \frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left\lbrack {{\widehat{\psi }}_{0}\left( k\right) {e}^{-{i\omega }\left( k\right) t}}\right\rbrack {dk}. \] By the uniqueness of the Fourier decomposition (i.e., the injectivity of the inverse Fourier... | Yes |
Theorem 4.5 Suppose \( {\psi }_{0} \in {L}^{2}\left( \mathbb{R}\right) \cap {L}^{1}\left( \mathbb{R}\right) \) . Then \( \psi \left( {x, t}\right) \), as defined by (4.5), may be computed for all \( t \neq 0 \) as\n\n\[ \psi \left( {x, t}\right) = \sqrt{\frac{m}{{2\pi it}\hslash }}{\int }_{-\infty }^{\infty }\exp \left... | The expression for \( \psi \left( {x, t}\right) \) is \( {\left( 2\pi \right) }^{-1/2}{K}_{t} * {\psi }_{0} \), where \( {K}_{t} \) is as in (4.8).\n\nProof. For any set \( E \subset \mathbb{R} \), let \( {1}_{E} \) denote the indicator function of \( E \), that is, the function that is 1 on \( E \) and 0 elsewhere. Th... | No |
A solution to the approximate equations (4.13) and (4.16) with initial condition \( \theta \left( {x,0}\right) = {p}_{0}x/\hslash \) is given by\n\n\[ \theta \left( {x, t}\right) = \frac{{p}_{0}}{\hslash }\left( {x - \frac{{p}_{0}}{2m}t}\right) \]\n\n(4.17)\n\nand\n\n\[ A\left( {x, t}\right) = {A}_{0}\left( {x - \frac{... | Proof. Although (4.16) is a nonlinear equation, we can find a solution to it with the simple initial conditions \( \theta \left( {x,0}\right) = {p}_{0}x/\hslash \), namely,\n\n\[ \theta \left( {x, t}\right) = \frac{{p}_{0}x}{\hslash } - \frac{{p}_{0}^{2}}{{2m}\hslash }t \]\n\n\[ = \frac{{p}_{0}}{\hslash }\left( {x - \f... | Yes |
Proposition 4.8 The speed at which a pure exponential solution of the free Schrödinger equation propagates is\n\n\[ \text{ phase velocity } = \frac{\omega \left( {k}_{0}\right) }{{k}_{0}} = \frac{\hslash {k}_{0}}{2m} = \frac{{p}_{0}}{2m}. \] | By contrast, the (approximate) speed at which the wave packet propagates is\n\n\[ \text{ group velocity } = {\left. \frac{d\omega }{dk}\right| }_{k = {k}_{0}} = \frac{\hslash {k}_{0}}{m} = \frac{{p}_{0}}{m}. \] | Yes |
Proposition 4.9 Let \( \psi \left( {x, t}\right) \) be the exact solution to the free Schrödinger equation with initial condition \( {\psi }_{0} \), and let \( \phi \left( {x, t}\right) \) be the approximate solution given by the right-hand side of (4.25). Then the following \( {L}^{2} \) estimate holds:\n\n\[ \paralle... | Proof. Let \( \widehat{\psi }\left( {k, t}\right) \) and \( \widehat{\phi }\left( {k, t}\right) \) denote the Fourier transforms of \( \phi \) and \( \psi \) with respect to \( x \), with \( t \) fixed. From (4.22) we can read off that\n\n\[ \widehat{\psi }\left( {k, t}\right) = {e}^{-{i\omega }\left( k\right) t}{\wide... | Yes |
For a wave function \( \psi \left( {x, t}\right) \) evolving according to the free Schrödinger equation on \( {\mathbb{R}}^{1} \), the expectation values for \( X \) and \( {X}^{2} \) evolve as follows:\n\n\[ \langle X{\rangle }_{\psi \left( t\right) } = \langle X{\rangle }_{{\psi }_{0}} + \frac{t}{m}\langle P{\rangle ... | Proof. We compute that\n\n\[ \left\lbrack {{P}^{2}, X}\right\rbrack = {P}^{2}X - {PXP} + {PXP} - X{P}^{2} \]\n\n\[ = P\left\lbrack {P, X}\right\rbrack + \left\lbrack {P, X}\right\rbrack P \]\n\n\[ = - {2i}\hslash P\text{.} \]\n\nThus (as we have already noted in Sect. 3.7.5),\n\n\[ \frac{d}{dt}\langle X{\rangle }_{\psi... | Yes |
Proposition 5.1 Suppose \( \psi \) is smooth on each of the intervals \( \left( {-\infty , - A}\right) \) , \( \left( {-A, A}\right) \), and \( \left( {A,\infty }\right) \) . Then \( \psi \) belongs to the domain of \( \widehat{H} \) [with potential function given by (5.1)] if and only if the (1) \( \psi \) and \( {d\p... | Proof. Suppose first that \( \psi \) satisfies the conditions (1) and (2). Then it is not hard to see (Exercise 1) that the second derivative of \( \psi \) in the distribution sense is simply the function \( {d}^{2}\psi /d{x}^{2} \), computed in the ordinary pointwise sense for \( x \neq \pm A \) . (The second derivati... | No |
Proposition 5.2 Let \( \psi \) be the function defined in (5.6)-(5.8). Then there exist nonzero constants \( a \) and \( b \) so that \( \psi \) belongs to the domain of \( \widehat{H} \) if and only if the following matching condition holds:\n\n\[ \sqrt{\varepsilon } = \sqrt{c - \varepsilon }\tan \left( {\sqrt{c - \va... | Proof. Clearly both \( \psi \) and \( {d}^{2}\psi /d{x}^{2} \) belong to \( {L}^{2}\left( \mathbb{R}\right) \) . Thus, in light of Proposition 5.1, we need only ensure that \( \psi \left( x\right) \) and \( {\psi }^{\prime }\left( x\right) \) are continuous at \( x = \pm A \) . Since the exponential functions are never... | Yes |
Proposition 5.3 For all positive values of \( c \) and \( A \), there exists at least one \( \varepsilon \in \left( {0, c}\right) \) such that (5.9) holds. | Proof. Case 1: \( \sqrt{c}A < \pi /2 \) . In this case, as \( \varepsilon \) varies between 0 and \( c \) , the left-hand side of (5.9) will vary between 0 and some positive number, whereas the right-hand side of (5.9) will vary between some positive number and 0 . By the intermediate value theorem, there must exist \(... | Yes |
Proposition 5.4 For any positive values of \( A \) and \( C \), there exists at least one value of \( E \) in the range \( - C < E < 0 \) for which (5.2) has a nonzero solution in the domain of \( \widehat{H} \), given by the formula\n\n\[ \n\psi \left( x\right) = \left\{ {\\begin{array}{ll} \cos \left( {\\sqrt{c - \\v... | In Proposition 5.4, we have not normalized \( \\psi \) to be a unit vector in \( {L}^{2}\\left( \\mathbb{R}\\right) \), but rather have normalized \( \\psi \) to equal 1 at the origin. In Figs. 5.5- 5.7, we plot our eigenfunction in several different cases. In Fig. 5.5, we have a \ | No |
Proposition 7.2 For all \( A \in \mathcal{B}\left( \mathbf{H}\right) \), we have\n\n\[ \begin{Vmatrix}{A}^{ * }\end{Vmatrix} = \parallel A\parallel \]\n\nand\n\n\[ \begin{Vmatrix}{{A}^{ * }A}\end{Vmatrix} = \parallel A{\parallel }^{2}. \] | Proof. The operator norm of \( A \) can also be computed as\n\n\[ \parallel A\parallel = \mathop{\sup }\limits_{{\parallel \psi \parallel = 1}}\parallel {A\psi }\parallel \]\n\nFurthermore, for any vector \( \phi \in \mathbf{H},\parallel \phi \parallel = \mathop{\sup }\limits_{{\parallel \chi \parallel = 1}}\left| {\la... | Yes |
Proposition 7.3 For all \( A \in \mathcal{B}\left( \mathbf{H}\right) \), we have\n\n\[{\left\lbrack \operatorname{Range}\left( A\right) \right\rbrack }^{ \bot } = \ker \left( {A}^{ * }\right)\]\n\nwhere for any \( B \in \mathcal{B}\left( \mathbf{H}\right) ,\ker \left( B\right) \) denotes the kernel of \( B \) . | Proof. Suppose first that \( \psi \) belongs to \( {\left\lbrack \operatorname{Range}\left( A\right) \right\rbrack }^{ \bot } \) . Then for all \( \phi \in \mathbf{H} \) , we have\n\n\[0 = \langle \psi ,{A\phi }\rangle = \left\langle {{A}^{ * }\psi ,\phi }\right\rangle\]\n\n(7.4)\n\nThis implies that \( {A}^{ * }\psi =... | Yes |
Lemma 7.6 Suppose \( X \in \mathcal{B}\left( \mathbf{H}\right) \) satisfies \( \parallel X\parallel < 1 \) . Then the operator \( I - X \) is invertible, with the inverse given by the following convergent series in \( \mathcal{B}\left( \mathbf{H}\right) \) : | Proof. As a consequence of (7.2), we have \( \begin{Vmatrix}{X}^{m}\end{Vmatrix} \leq \parallel X{\parallel }^{m} \) . The (geometric) series on the right-hand side of (7.5) is therefore absolutely convergent and thus convergent in the Banach space \( \mathcal{B}\left( \mathbf{H}\right) \) (Appendix A.3.4). If we multi... | Yes |
Lemma 7.8 If \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is self-adjoint, then for all \( \lambda = a + {ib} \in \mathbb{C} \), we have\n\n\[ \langle \left( {A - {\lambda I}}\right) \psi ,\left( {A - {\lambda I}}\right) \psi \rangle \geq {b}^{2}\langle \psi ,\psi \rangle \] | Proof. We compute that\n\n\[ \langle \left( {A - \left( {a + {ib}}\right) I}\right) \psi ,\left( {A - \left( {a + {ib}}\right) I}\right) \psi \rangle \]\n\n\[ = \langle \left( {A - {aI}}\right) \psi ,\left( {A - {aI}}\right) \psi \rangle + {ib}\langle \psi ,\left( {A - {aI}}\right) \psi \rangle \]\n\n\[ - {ib}\langle \... | Yes |
Let \( \mathbf{H} = {L}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) \) and let \( A \) be the operator on \( \mathbf{H} \) defined \( {by} \)\n\n\[ \left( {A\psi }\right) \left( x\right) = {x\psi }\left( x\right) \]\n\nThen this operator is bounded and self-adjoint, and its spectrum is given by\n\n\[ \sigma \left... | Proof. It is apparent that \( \parallel {A\psi }\parallel \leq \parallel \psi \parallel \) and that \( \langle \phi ,{A\psi }\rangle = \langle {A\phi },\psi \rangle \) for all \( \phi ,\psi \in \mathbf{H} \), so that \( A \) is bounded and self-adjoint. Given \( \lambda \in \left( {0,1}\right) \), consider the function... | Yes |
Proposition 7.11 (Operator-Valued Integration) Let \( \Omega \) be a \( \sigma \) -algebra in a set \( X \) and let \( \mu : \Omega \rightarrow \mathcal{B}\left( \mathbf{H}\right) \) be a projection-valued measure. Then there exists a unique linear map, denoted \( f \mapsto {\int }_{\Omega }{fd\mu } \), from the space ... | Proof of Proposition 7.11. Given a projection-valued measure \( \mu \) and a bounded measurable function \( f \) on \( X \), define a map \( {Q}_{f} : \mathbf{H} \rightarrow \mathbb{C} \) by\n\n\[ {Q}_{f}\left( \psi \right) = {\int }_{X}{fd}{\mu }_{\psi } \]\n\nwhere \( {\mu }_{\psi } \) is given by (7.14). If \( f \) ... | Yes |
Theorem 7.12 (Spectral Theorem, First Form) If \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is selfadjoint, then there exists a unique projection-valued measure \( {\mu }^{A} \) on the Borel \( \sigma \) -algebra in \( \sigma \left( A\right) \), with values in projections on \( \mathbf{H} \), such that\n\n\[{\int }_... | Since the spectrum \( \sigma \left( A\right) \) of \( A \) is bounded, the function \( f\left( \lambda \right) \mathrel{\text{:=}} \lambda \) is bounded on \( \sigma \left( A\right) \) . The proof of this theorem is given in Chap. 8. | No |
Proposition 7.17 Suppose \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is self-adjoint and \( \psi \in \mathbf{H} \) is a unit vector. Then there exists a unique probability measure \( {\mu }_{\psi }^{A} \) on \( \mathbb{R} \) such that\n\n\[{\int }_{\mathbb{R}}{\lambda }^{m}d{\mu }_{\psi }^{A}\left( \lambda \right) ... | Proof. We define a measure \( {\mu }_{\psi }^{A} \) on \( \sigma \left( A\right) \) as in Sect. 7.2.2 by\n\n\[{\mu }_{\psi }^{A}\left( E\right) = \left\langle {\psi ,{\mu }^{A}\left( E\right) \psi }\right\rangle\]\n\nThe properties of integration with respect to \( {\mu }^{A} \) then tell us that\n\n\[\left\langle {\ps... | Yes |
Proposition 7.23 Suppose \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is self-adjoint, \( {\mu }^{A} \) is the projection-valued measure given by Theorem 7.12 and \( \mu \) is a real-valued measure as in Theorem 7.19. If \( \dim {\mathbf{H}}_{\lambda } > 0 \) for \( \mu \) -almost every \( \lambda \), then for any B... | Proof. As we have remarked, given a direct integral as in Theorem 7.19, we can construct a projection-valued measure by means of (7.21), and this projection-valued measure satisfies \( {\int }_{\sigma \left( A\right) }{\lambda d}{\mu }^{A}\left( \lambda \right) = A \) . This projection-valued measure must coincide with... | Yes |
Proposition 7.24 Suppose \( {A}_{1} \) and \( {A}_{2} \) are bounded self-adjoint operators on separable Hilbert spaces \( {\mathbf{H}}_{1} \) and \( {\mathbf{H}}_{2} \), respectively. Choose direct integral representations for \( {A}_{1} \) and \( {A}_{2} \) as in Theorem 7.19, with the associated measures \( {\mu }_{... | See Exercise 12 for a proof of this result. | No |
If \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is self-adjoint, the norm and the spectral radius of \( A \) are equal: | Proof of Lemma 8.1. We know that \( R\left( A\right) \leq \parallel A\parallel \) . To show that \( R\left( A\right) = \) \( \parallel A\parallel \), we wish to argue that \( {\left( A - \lambda I\right) }^{-1} \) is a holomorphic operator-valued function of \( \lambda \) on the set \( \left| \lambda \right| > R\left( ... | Yes |
Lemma 8.2 (Spectral Mapping Theorem) For all \( A \in \mathcal{B}\left( \mathbf{H}\right) \) and all polynomials \( p \), we have\n\n\[ \sigma \left( {p\left( A\right) }\right) = p\left( {\sigma \left( A\right) }\right) . \]\n\nThat is to say, the spectrum of \( p\left( A\right) \) consists precisely of the numbers of ... | Proof. The result is trivial if \( p \) is constant. When \( \deg p \geq 1 \), let \( p \) given by\n\n\[ p\left( z\right) = {a}_{n}{z}^{n} + {a}_{n - 1}{z}^{n - 1} + \cdots + {a}_{0} \]\n\nbe an arbitrary polynomial. We first show that \( p\left( {\sigma \left( A\right) }\right) \subset \sigma \left( {p\left( A\right)... | Yes |
Proposition 8.3 Suppose \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is self-adjoint. Then there exists a unique bounded linear map from \( \mathcal{C}\left( {\sigma \left( A\right) ;\mathbb{R}}\right) \) into \( \mathcal{B}\left( \mathbf{H}\right) \), denoted by \( f \mapsto \) \( f\left( A\right) \), such that whe... | Proof. Note that if \( A \) is self-adjoint, then \( p\left( A\right) \) is self-adjoint provided that \( p \) is a real-valued polynomial (i.e., one where all the coefficients are real numbers). Thus, combining the spectral mapping theorem with the equality of the norm and spectral radius, we have the following: If \(... | Yes |
Proposition 8.4 If \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is self-adjoint, the (real-valued) continuous functional calculus for \( A \), mapping \( \mathcal{C}\left( {\sigma \left( A\right) ;\mathbb{R}}\right) \) into \( \mathcal{B}\left( \mathbf{H}\right) \), has the following properties.\n\n1. Multiplicativi... | Proof. Point 1 holds for polynomials and thus, by taking limits, for all \( f \in C\left( {\sigma \left( A\right) ;\mathbb{R}}\right) \) . Furthermore, if \( p \) is a real-valued polynomial and \( A \) is self-adjoint, then \( p\left( A\right) \) is self-adjoint. From this, we get Point 2 by taking limits. If \( f \in... | Yes |
Theorem 8.5 (Riesz Representation Theorem) Let \( X \) be a compact metric space and let \( \mathcal{C}\left( {X;\mathbb{R}}\right) \) denote the space of continuous, real-valued functions on \( X \) . Suppose \( \Lambda : \mathcal{C}\left( {X;\mathbb{R}}\right) \rightarrow \mathbb{R} \) is a linear functional with the... | See pp. 353-354 of Volume I of [34] for a short proof in the case in which \( X \) is a compact subset of \( \mathbb{R} \), which is all we really require. For the full result stated above, see Theorems 7.2 and 7.8 in [12]. | No |
Proposition 8.7 For any bounded measurable function \( f \) on \( \sigma \left( A\right) \), the map \( {Q}_{f} \) in Definition 8.6 is a bounded quadratic form. | Proof. Let \( \mathcal{F} \) denote the space of all bounded, Borel-measurable functions \( f \) for which \( {Q}_{f} \) is a quadratic form. Then \( \mathcal{F} \) is a vector space and contains \( \mathcal{C}\left( {\sigma \left( A\right) ;\mathbb{R}}\right) \). Furthermore, \( \mathcal{F} \) is closed under uniforml... | "No" |
Proposition 8.9 For any two bounded measurable functions \( f \) and \( g \), we have\n\n\[ \left( {fg}\right) \left( A\right) = f\left( A\right) g\left( A\right) \] | Proof. Let \( {\mathcal{F}}_{1} \) denote the space of bounded measurable functions \( f \) such that \( \left( {fg}\right) \left( A\right) = f\left( A\right) g\left( A\right) \) for all \( g \in \mathcal{C}\left( {\sigma \left( A\right) ;\mathbb{R}}\right) \) . Then \( {\mathcal{F}}_{1} \) is a vector space and contai... | No |
Lemma 8.11 Suppose \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is self-adjoint and \( \psi \) is a cyclic vector for \( A \) . Let \( {\mu }_{\psi } \) be the unique measure on \( \sigma \left( A\right) \), given by Theorem 8.5, for which\n\n\[ \langle \psi, f\left( A\right) \psi \rangle = {\int }_{\sigma \left( A\... | Proof. We start by defining \( U \) on the complex vector space of vectors of the form \( p\left( A\right) \psi \), where \( p \) is a complex-valued polynomial, as follows:\n\n\[ U\left\lbrack {p\left( A\right) \psi }\right\rbrack = p. \]\n\nTo show that \( U \) is well defined, write \( p \) as \( p = {p}_{1} + i{p}_... | Yes |
Lemma 8.12 Suppose \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is self-adjoint and \( {\mu }^{A} \) is the associated projection-valued measure on \( \sigma \left( A\right) \), as in Theorem 8.10. Then there exists a non-negative real-valued measure \( \mu \) on \( \sigma \left( A\right) \) such that for all Borel ... | Proof. Let \( \left\{ {e}_{j}\right\} \) be an orthonormal basis for \( \mathbf{H} \) and let \( {\mu }_{{e}_{j}} \) be the associated real-valued measures, given by \( {\mu }_{{e}_{j}}\left( E\right) = \left\langle {{e}_{j},{\mu }^{A}\left( E\right) {e}_{j}}\right\rangle \) . Then \( {\mu }_{{e}_{j}}\left( {\sigma \le... | Yes |
Lemma 8.13 If \( A \in \mathcal{B}\left( \mathbf{H}\right) \) is self-adjoint, then \( \mathbf{H} \) can be decomposed as an orthogonal direct sum of closed nonzero subspaces \( {W}_{j} \), where each \( {W}_{j} \) is invariant under \( A \) and where the restriction of \( A \) to \( {W}_{j} \) has a cyclic vector \( {... | Proof. Recall our standing assumption that \( \mathbf{H} \) is separable, and let \( \left\{ {\phi }_{j}\right\} \) be a countable dense subset of \( \mathbf{H} \) . Let \( {W}_{1} \) be the closed subspace of \( \mathbf{H} \) spanned by \( {\phi }_{1}, A{\phi }_{1},{A}^{2}{\phi }_{1},\ldots \) Then \( {W}_{1} \) is in... | Yes |
Proposition 9.4 An unbounded operator \( A \) is symmetric if and only if \( {A}^{ * } \) is an extension of \( A \) . | Proof. If \( A \) is symmetric, then for all \( \phi \in \operatorname{Dom}\left( A\right) ,\left( {9.1}\right) \) and the Cauchy-Schwarz inequality show that\n\n\[ \left| {\langle \phi ,{A\psi }\rangle }\right| \leq \parallel {A\phi }\parallel \parallel \psi \parallel \]\n\nshowing that \( \phi \in \operatorname{Dom}\... | Yes |
Proposition 9.8 1. If \( A \) is an unbounded operator on \( \mathbf{H} \), then the graph of the operator \( {A}^{ * } \) (which may or may not be densely defined) is closed in \( \mathbf{H} \times \mathbf{H} \) . | Proof. Suppose \( {\psi }_{n} \) is a sequence in the domain of \( {A}^{ * } \) that converges to some \( \psi \in \mathbf{H} \) . Suppose also that \( {A}^{ * }{\psi }_{n} \) converges to some \( \phi \in \mathbf{H} \) . Then \( \left\langle {{\psi }_{n}, A \cdot }\right\rangle = \left\langle {{A}^{ * }{\psi }_{n}, \c... | Yes |
Corollary 9.9 If \( A \) is a symmetric operator with \( \operatorname{Dom}\left( A\right) = \mathbf{H} \), then \( A \) is bounded. | Proof. Since \( A \) is symmetric, it is closable by Proposition 9.8. But since the domain of \( A \) is already all of \( \mathbf{H} \), the closure of \( A \) must coincide with \( A \) itself. (The closure of \( A \) always agrees with \( A \) on \( \operatorname{Dom}\left( A\right) \), which in this case is all of ... | Yes |
Proposition 9.10 If \( A \) is a closable operator on \( \mathbf{H} \), then the adjoint of \( {A}^{cl} \) coincides with the adjoint of \( A \) . | Proof. Suppose that for some \( \psi \in \mathbf{H} \) there exists a \( \phi \) such that \( \left\langle {\psi ,{A}^{cl}\chi }\right\rangle = \) \( \langle \phi ,\chi \rangle \) for all \( \chi \in \operatorname{Dom}\left( {A}^{cl}\right) \) . Since \( {A}^{cl} \) is an extension of \( A \), it follows that \( \langl... | Yes |
Proposition 9.11 If \( A \) is essentially self-adjoint, then \( {A}^{cl} \) is the unique self-adjoint extension of \( A \) . | Proof. Suppose \( B \) is a self-adjoint extension of \( A \) . Since \( B = {B}^{ * }, B \) is closed and is, therefore, an extension of \( {A}^{cl} \) . It then follows from the definition of the adjoint that \( \operatorname{Dom}\left( {B}^{ * }\right) \subset \operatorname{Dom}\left( {A}^{cl}\right) \) . Thus, we h... | Yes |
Proposition 9.12 If \( A \) is an unbounded operator on \( \mathbf{H} \), then\n\n\[{\left( \operatorname{Range}\left( A\right) \right) }^{ \bot } = \ker \left( {A}^{ * }\right)\] | Proof. First assume that \( \psi \in {\left( \operatorname{Range}\left( A\right) \right) }^{ \bot } \) . Then for all \( \phi \in \operatorname{Dom}\left( A\right) \) we have\n\n\[ \langle \psi ,{A\phi }\rangle = 0 \]\n\nThat is to say, the linear functional \( \langle \psi, A \cdot \rangle \) is bounded-in fact, zero-... | Yes |
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