id stringlengths 40 40 | title stringlengths 15 120 | text stringlengths 41 3.14k | source stringclasses 2
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d9ce0632cd208dd069cb2397a0a13b4e45e33515 | the spherical tensor formalism provides a common platform | the spherical tensor formalism provides a common platform for treating coherence and relaxation in nuclear magnetic resonance. in nmr and epr, spherical tensor operators are employed to express the quantum dynamics of particle spin, by means of an equation of motion for the density matrix entries, or to formulate dynam... | wikipedia |
1694df5c3004970be3d3840b0f613613043096b0 | the selection rule m β² = q + | the selection rule m β² = q + m in the clebschβgordan coefficient means that many of the integrals vanish, so we have exaggerated the total number of integrals that need to be done. but had we worked with the cartesian components r of r, this selection rule might not have been obvious. in any case, even with the selecti... | wikipedia |
51a5caea5daa393e1da282a3577470b2277c1e5a | the radial integral is independent of the three | the radial integral is independent of the three magnetic quantum numbers (m β², q, m), and the trick we have just used does not help us to evaluate it. but it is only one integral, and after it has been done, all the other integrals can be evaluated just by computing or looking up clebschβgordan coefficients. | wikipedia |
89705d45ffb7dbe4e7746ecc5bb09a09e0146687 | we see that all the dependence on the | we see that all the dependence on the three magnetic quantum numbers (mβ²,q,m) is contained in the angular part of the integral. moreover, the angular integral can be evaluated by the three- y formula, whereupon it becomes proportional to the clebsch-gordan coefficient, | wikipedia |
101d2b19a91ec0d6e27471b46759ee03a329ff1c | for q = 1, 0, β1, where q | for q = 1, 0, β1, where q appears explicitly as a magnetic quantum number. this equation reveals a relationship between vector operators and the angular momentum value β = 1, something we will have more to say about presently. now the matrix elements become a product of a radial integral times an angular integral, β¨ n ... | wikipedia |
d4511d51a45f738c7a38cda440073946163de470 | r y 11 (ΞΈ, Ο) = β r | r y 11 (ΞΈ, Ο) = β r 3 8 Ο sin (ΞΈ) e i Ο = 3 4 Ο (β x + i y 2) r y 10 (ΞΈ, Ο) = r 3 4 Ο cos (ΞΈ) = 3 4 Ο z r y 1 β 1 (ΞΈ, Ο) = r 3 8 Ο sin (ΞΈ) e β i Ο = 3 4 Ο (x β i y 2) {\displaystyle {\begin{aligned}ry_{11}(\theta,\phi)&=&&-r{\sqrt {\frac {3}{8\pi }}}\sin(\theta)e^{i\phi }&=&{\sqrt {\frac {3}{4\pi }}}\left(-{\frac {x+iy... | wikipedia |
d3a14c2ee786b1e0d5d6e2a9e46b40b61ad1447f | where, the initial state is on the right | where, the initial state is on the right and the final one on the left. the position operator r has three components, and the initial and final levels consist of 2β + 1 and 2ββ² + 1 degenerate states, respectively. therefore if we wish to evaluate the intensity of a spectral line as it would be observed, we really have ... | wikipedia |
9c021ff41f9555c664af3167c3280a8a4beb2320 | Ο n β m (r, ΞΈ, Ο) = | Ο n β m (r, ΞΈ, Ο) = r n β (r) y β m (ΞΈ, Ο) {\displaystyle \psi _{n\ell m}(r,\theta,\phi)=r_{n\ell }(r)y_{\ell m}(\theta,\phi)} | wikipedia |
1fbf3bf222c899f6692e390ab04bfe474eac5ca7 | the transition amplitude is proportional to matrix elements | the transition amplitude is proportional to matrix elements of the dipole operator between the initial and final states. we use an electrostatic, spinless model for the atom and we consider the transition from the initial energy level e to final level e. these levels are degenerate, since the energy does not depend on ... | wikipedia |
0fe23ab8de9de1e5e7784c2d8ed99cd629933a9b | spherical tensors can also be formed from algebraic | spherical tensors can also be formed from algebraic combinations of the spin operators s, s, s, as matrices, for a spin system with total quantum number j = β + s (and β = 0). spin operators have the ladder operators: | wikipedia |
8e70eedc4d856241e8dd4522df4f4dff7e08742b | the hermitian adjoint of a spherical tensor may | the hermitian adjoint of a spherical tensor may be defined as (t β ) q (k) = (β 1) k β q (t β q (k)) β . {\displaystyle (t^{\dagger })_{q}^{(k)}=(-1)^{k-q}(t_{-q}^{(k)})^{\dagger }.} there is some arbitrariness in the choice of the phase factor: any factor containing (β1) will satisfy the commutation relations. the above... | wikipedia |
2ee812c6dd55459491a8d366a81c54596d68cbf3 | define an operator by its spectrum: Ξ₯ l | define an operator by its spectrum: Ξ₯ l m | r β© = r l y l m (ΞΈ, Ο) | r β© = Ξ₯ l m (r β) | r β© {\displaystyle \upsilon _{l}^{m}|r\rangle =r^{l}y_{l}^{m}(\theta,\phi)|r\rangle =\upsilon _{l}^{m}({\vec {r}})|r\rangle } since for spherical harmonics under rotation: y β = k m = q (n) = β¨ n | k, q β© β u (r) β y β = k m = q (n... | wikipedia |
73bc96f99cb1ae10d970881a38adc83b4633a4bc | define an operator by its spectrum: Ξ₯ l | β) u (r) = β m β² d m β², m (l) (r β 1) Ξ₯ l m β² (r β) {\displaystyle \upsilon _{l}^{m}({\vec {r}})\rightarrow u(r)^{\dagger }\upsilon _{l}^{m}({\vec {r}})u(r)=\sum _{m'}d_{m',m}^{(l)}(r^{-1})\upsilon _{l}^{m'}({\vec {r}})} then Ξ₯ l m (v β) {\displaystyle \upsilon _{l}^{m}({\vec {v}})}, where v β {\displaystyle {\vec {v}... | wikipedia |
10fd2054195c7108c071dce7e7d01d2561c331b3 | using the infinitesimal rotation operator and its hermitian | using the infinitesimal rotation operator and its hermitian conjugate, one can derive the commutation relation in the spherical basis: = β q β² d (j a) q q β² (2) t ^ q β² (2) = β q β² β¨ j = 2, m = q | j a | j = 2, m = q β² β© t ^ q β² (2) {\displaystyle \left=\sum _{q'}{d(j_{a})}_{qq'}^{(2)}{\widehat {t}}_{q'}^{(2)}=\sum _{q... | wikipedia |
3bde26f6812e1df607a8c5c00260015b07719b67 | t ^ Β± 2 (2) = a ^ | t ^ Β± 2 (2) = a ^ Β± 1 b ^ Β± 1 t ^ Β± 1 (2) = 1 2 (a ^ Β± 1 b ^ 0 + a ^ 0 b ^ Β± 1) t ^ 0 (2) = 1 6 (a ^ + 1 b ^ β 1 + a ^ β 1 b ^ + 1 + 2 a ^ 0 b ^ 0) {\displaystyle {\begin{aligned}{\widehat {t}}_{\pm 2}^{(2)}&={\widehat {a}}_{\pm 1}{\widehat {b}}_{\pm 1}\\{\widehat {t}}_{\pm 1}^{(2)}&={\tfrac {1}{\sqrt {2}}}\left({\wide... | wikipedia |
0a4a59f4aa40729b7f7cafc4dccc45aad2cdb414 | combination of two spherical tensors a q 1 | combination of two spherical tensors a q 1 (k 1) {\displaystyle a_{q_{1}}^{(k_{1})}} and b q 2 (k 2) {\displaystyle b_{q_{2}}^{(k_{2})}} in the following manner involving the clebschβgordan coefficients can be proved to give another spherical tensor of the form: t q (k) = β q 1, q 2 β¨ k 1, k 2; q 1, q 2 | k 1, k 2; k, ... | wikipedia |
cd7fffe6ec5b3fd0147942666c5d61b797c2b82c | in the following section, construction of spherical tensors | in the following section, construction of spherical tensors will be discussed. for example, since example of spherical vector operators is shown, it can be used to construct higher order spherical tensor operators. in general, spherical tensor operators can be constructed from two perspectives. one way is to specify ho... | wikipedia |
00f3b68ab322f9cecd5300835166587da91db905 | where the exponential form is given by bakerβhausdorff | where the exponential form is given by bakerβhausdorff lemma. hence, the above commutation relations and the transformation property are equivalent definitions of spherical tensor operators. it can also be shown that { a d j ^ i } {\displaystyle \{ad_{{\hat {j}}_{i}}\}} transform like a vector due to their commutation ... | wikipedia |
f6fd3aebcec25d8379e50252c0488e1027a29ad2 | t ^ m (j) β u (r) β | t ^ m (j) β u (r) β t ^ m (j) u (r) = e x p (i ΞΈ β n ^ β
a d j β) t ^ m (j) = β m β² d m β² m (j) (r β 1) t ^ m β² (j) {\displaystyle {\widehat {t}}_{m}^{(j)}\rightarrow u(r)^{\dagger }{\widehat {t}}_{m}^{(j)}u(r)=exp\left({i{\frac {\theta }{\hbar }}{\hat {n}}\cdot ad_{\vec {j}}}\right){\widehat {t}}_{m}^{(j)}=\sum _{m'}d... | wikipedia |
bde5c2faea58d1c6d14a4a8dd3fad94786cf88c0 | we find due to similarity of actions of | we find due to similarity of actions of j {\displaystyle j} on wavefunction | j, m β© {\displaystyle |j,m\rangle } and the commutation relations on t ^ m (j) {\displaystyle {\widehat {t}}_{m}^{(j)}}, that: | wikipedia |
bf8bc97528891a7482aee81870b2dc4e5b83d842 | if, only the commutation relations hold, using the | if, only the commutation relations hold, using the following relation, | j, m β© β u (r) | j, m β© = e x p (β i ΞΈ β n ^ β
j β) | j, m β© = β m β² d m β² m (j) (r) | j, m β² β© {\displaystyle |j,m\rangle \rightarrow u(r)|j,m\rangle =exp\left(-{i{\frac {\theta }{\hbar }}{\hat {n}}\cdot {\vec {j}}}\right)|j,m\rangle =\sum _{m'}d... | wikipedia |
27f3d09ee59a504355fd6844dc6df62f07d0fd41 | for choices of n ^ = x ^ | for choices of n ^ = x ^ Β± i y ^ {\displaystyle {\hat {n}}={\hat {x}}\pm i{\hat {y}}} or n ^ = z ^ {\displaystyle {\hat {n}}={\hat {z}}}, we get: = β (j β m) (j Β± m + 1) t ^ m Β± 1 (j) = β m t ^ m (j) {\displaystyle {\begin{aligned}\left&=\hbar {\sqrt {(j\mp m)(j\pm m+1)}}{\widehat {t}}_{m\pm 1}^{(j)}\\\left&=\hbar m{\w... | wikipedia |
3bfe6ad358e42acc91defe5c8b00ea722bae026c | = β m β² t ^ m β² | = β m β² t ^ m β² (j) β¨ j, m β² | j β β
n ^ | j, m β© {\displaystyle {=\sum _{m'}{\widehat {t}}_{m'}^{(j)}\langle j,m'|{\vec {j}}\cdot {\hat {n}}|j,m\rangle } | wikipedia |
34377b77ceadfc3f1570df9e1e7b68ce7fcb43ab | u (r) β t ^ m (j) u | u (r) β t ^ m (j) u (r) = (1 + i Ο΅ n ^ β
j β β + o (Ο΅ 2)) t ^ m (j) (1 β i Ο΅ n ^ β
j β β + o (Ο΅ 2)) = β m β² β¨ j, m β² | (1 + i Ο΅ n ^ β
j β β + o (Ο΅ 2)) | j, m β© t ^ m β² (j) {\displaystyle u(r)^{\dagger }{\widehat {t}}_{m}^{(j)}u(r)=\left(1+{\frac {i\epsilon {\hat {n}}\cdot {\vec {j}}}{\hbar }}+{\mathcal {o}}(\epsilon ^{... | wikipedia |
1e5be55eb924c7c2de6275af6853922560b4767a | t ^ m (j) β u (r) β | t ^ m (j) β u (r) β t ^ m (j) u (r) = β m β² d m β² m (j) (r β 1) t ^ m β² (j) {\displaystyle {\widehat {t}}_{m}^{(j)}\rightarrow u(r)^{\dagger }{\widehat {t}}_{m}^{(j)}u(r)=\sum _{m'}d_{m'm}^{(j)}(r^{-1}){\widehat {t}}_{m'}^{(j)}} | wikipedia |
30150ea780a70b9e33216e76f20802c5591d2a9b | in general cartesian tensors of rank greater than | in general cartesian tensors of rank greater than 1 are reducible. in quantum mechanics, this particular example bears resemblance to the addition of two spin one particles where both are 3 dimensional, hence the total space being 9 dimensional, can be formed by spin 0, spin 1 and spin 2 systems each having 1 dimension... | wikipedia |
f0a702ccafdfd950be0ad1cdd30b5e8090fbbcfe | where: t ^ i j (0) = v | where: t ^ i j (0) = v ^ k w ^ k 3 Ξ΄ i j {\displaystyle {\widehat {t}}_{ij}^{(0)}={\frac {{\widehat {v}}_{k}{\widehat {w}}_{k}}{3}}\delta _{ij}} t ^ i j (1) = 1 2 = v ^ {\displaystyle {\widehat {t}}_{ij}^{(1)}={\frac {1}{2}}\left={\widehat {v}}_{}} t ^ i j (2) = 1 2 (v ^ i w ^ j + v ^ j w ^ i) β 1 3 v ^ k w ^ k Ξ΄ i j =... | wikipedia |
00eb7e5dd64cb66cb0ac8a83e072a7b8ff1d13d0 | t ^ i j = 1 3 (v | t ^ i j = 1 3 (v β β
w β) Ξ΄ i j + (1 2 (v ^ i w ^ j β v ^ j w ^ i)) + (1 2 (v ^ i w ^ j + v ^ j w ^ i) β 1 3 (v β β
w β) Ξ΄ i j) = t (0) + t (1) + t (2) {\displaystyle {\hat {t}}_{ij}={\frac {1}{3}}({\vec {v}}\cdot {\vec {w}})\delta _{ij}+\left({\frac {1}{2}}({\hat {v}}_{i}{\hat {w}}_{j}-{\hat {v}}_{j}{\hat {w}}_{i})\ri... | wikipedia |
b6025b94c5af2665bcac20dc481b70d3177def1e | t ^ i j = 1 3 t | t ^ i j = 1 3 t ^ Ξ΄ i j + a ^ i j + s ^ i j {\displaystyle {\hat {t}}_{ij}={\frac {1}{3}}{\hat {t}}\delta _{ij}+{\hat {a}}_{ij}+{\hat {s}}_{ij}} | wikipedia |
db0d6a271e2f9cad19af468863eb0cbd0c8388be | from the above examples, the nine component { | from the above examples, the nine component { t ^ i j } {\displaystyle \{{\hat {t}}_{ij}\}} are split into subspaces formed by one, three and five components. these numbers add up to the number of components of the original tensor in a manner similar to the dimension of vector subspaces adding to the dimension of the s... | wikipedia |
5d0c3de2cde8f95b8d4c5605cf868066d40ccafe | if t ^ i j = v i | if t ^ i j = v i ^ w j ^ {\displaystyle {\hat {t}}_{ij}={\hat {v_{i}}}{\hat {w_{j}}}}, the invariant subspaces of { t ^ i j } {\displaystyle \{{\hat {t}}_{ij}\}} formed are represented by: | wikipedia |
df4750d6d21701131f6f161ca865bab01cdbbd4e | the subspace spanned by { t ^ i | the subspace spanned by { t ^ i j } {\displaystyle \{{\hat {t}}_{ij}\}} can be divided two subspaces; three independent antisymmetric components { a ^ i j } {\displaystyle \{{\hat {a}}_{ij}\}} and six independent symmetric component { s ^ i j } {\displaystyle \{{\hat {s}}_{ij}\}}, defined as a ^ i j = 1 2 (t ^ i j β t ... | wikipedia |
d8bf04e30715edc6deaa6081c138234c0172185c | we observe that the subspace spanned by linear | we observe that the subspace spanned by linear combinations of the rank two tensor components form an invariant subspace, ie. the subspace does not change under rotation since the transformed components itself is a linear combination of the tensor components. however, this subspace is not irreducible ie. it can be furt... | wikipedia |
6fac8c90cdcb58064c6014483736f06119aed463 | if v β {\displaystyle {\vec {v}}} and w | if v β {\displaystyle {\vec {v}}} and w β {\displaystyle {\vec {w}}} are two three dimensional vector operators, then a rank 2 cartesian dyadic tensors can be formed from nine operators of form t ^ i j = v i ^ w j ^ {\displaystyle {\hat {t}}_{ij}={\hat {v_{i}}}{\hat {w_{j}}}}, u (r) β t ^ i j u (r) = u (r) β (v i ^ w j... | wikipedia |
7321f95e26a0ebc5384729146e39aa9b9297f407 | if v β {\displaystyle {\vec {v}}} and w | }{\hat {t}}_{ij}u(r)=\sum _{k=1}^{3}\sum _{l=1}^{3}\left(r_{il}r_{jk}{\hat {t}}_{lk}\right)} the rhs of the equation is change of basis equation for twice contravariant tensors where the basis are transformed by r β 1 {\displaystyle r^{-1}} or the vector components transform by r {\displaystyle r} which matches transf... | wikipedia |
64d745c3ff49158c5c4187f2e1d5276d2445e114 | u (r) β t ^ p q r | u (r) β t ^ p q r β― u (r) = r p i r q j r r k β― t ^ i j k β― {\displaystyle u(r)^{\dagger }{\widehat {t}}_{pqr\cdots }u(r)=r_{pi}r_{qj}r_{rk}\cdots {\widehat {t}}_{ijk\cdots }} | wikipedia |
a230344f51ae0dbb30ea98aaf237b7500c10a308 | in general, a tensor operator is one that | in general, a tensor operator is one that transforms according to a tensor: u (r) β t ^ p q r β― a b c β― u (r) = r p, Ξ± r q, Ξ² r r, Ξ³ β― t ^ i j k β― Ξ± Ξ² Ξ³ β― r i, a β 1 r j, b β 1 r k, c β 1 β― {\displaystyle u(r)^{\dagger }{\widehat {t}}_{pqr\cdots }^{abc\cdots }u(r)=r_{p,\alpha }r_{q,\beta }r_{r,\gamma }\cdots {\widehat ... | wikipedia |
919e681bfd7dadc3ce5bc7d7bf69bc252490952c | the rotation transformation in the spherical basis (originally | the rotation transformation in the spherical basis (originally written in the cartesian basis) is then, due to similarity of commutation and operator shown above: u (r) β v ^ q u (r) = β q β² d q β² q (1) (r β 1) v ^ q β² {\displaystyle {u(r)}^{\dagger }{\widehat {v}}_{q}u(r)=\sum _{q'}{{d_{q'q}^{(1)}}(r^{-1})}{\widehat {... | wikipedia |
767233903e70f5b9841eaeed1ca1ca321f363792 | u (r) | j, k β© = exp | u (r) | j, k β© = exp (β i ΞΈ β n ^ β
j β) | j, k β© = β j β², k β² | j β², k β² β© β¨ j β², k β² | exp (β i ΞΈ β n ^ β
j β) | j, k β© = β k β² d k β² k (j) (r) | j, k β² β© {\displaystyle u(r)|j,k\rangle =\exp \left({-i{\frac {\theta }{\hbar }}{\hat {n}}\cdot {\vec {j}}}\right)|j,k\rangle =\sum _{j',k'}|j',k'\rangle \langle j',k'|\exp... | wikipedia |
210e2fe661924ce565f9758a9954a418df01eedf | u (r) | j, k β© = | | u (r) | j, k β© = | j, k β© β i ΞΈ β n ^ β
j β | j, k β© + β k = 2 β (β i ΞΈ β n ^ β
j β) k k ! | j, k β© = e x p (β i ΞΈ β n ^ β
j β) | j, k β© {\displaystyle u(r)|j,k\rangle =|j,k\rangle -i{\frac {\theta }{\hbar }}{\hat {n}}\cdot {\vec {j}}|j,k\rangle +\sum _{k=2}^{\infty }{\frac {\left(-i{\frac {\theta }{\hbar }}{\hat {n}}\... | wikipedia |
83ff029731dbc3ebc2db29dcc73445eae942ee04 | u (r) β v ^ q u (r) | u (r) β v ^ q u (r) = v ^ q + i ΞΈ β + β k = 2 β (i ΞΈ β) k k ! v ^ q = e x p (i ΞΈ β n ^ β
a d j β) v ^ q {\displaystyle {u(r)}^{\dagger }{\widehat {v}}_{q}u(r)={\widehat {v}}_{q}+i{\frac {\theta }{\hbar }}\left+\sum _{k=2}^{\infty }{\frac {\left(i{\frac {\theta }{\hbar }}\right)^{k}}{k!}}{\widehat {v}}_{q}=exp\left({i{\... | wikipedia |
2ed4444137410b0139af25a1ce43458f8ce4e42c | in the spherical basis, the generators of rotation | in the spherical basis, the generators of rotation are: j Β± 1 = β 1 2 j Β±, j 0 = j z {\displaystyle j_{\pm 1}=\mp {\frac {1}{\sqrt {2}}}j_{\pm }\,,\quad j_{0}=j_{z}} | wikipedia |
5dc6bf0bcaf19c1f2d362efd8f59c498c9b43c08 | which are of similar form of j z | which are of similar form of j z | 1, + 1 β© = + β | 1, + 1 β© j z | 1, 0 β© = 0 | 1, 0 β© j z | 1, β 1 β© = β β | 1, β 1 β© j + | 1, + 1 β© = 0 j + | 1, 0 β© = 2 β | 1, + 1 β© j + | 1, β 1 β© = 2 β | 1, 0 β© j β | 1, + 1 β© = 2 β | 1, 0 β© j β | 1, 0 β© = 2 β | 1, β 1 β© j β | 1, β 1 β© = 0 {\displaystyle {\begin{aligned}j_{z}|1,+1\r... | wikipedia |
9615de3030e0a56b42f0aa77f0f7be5b380e6f5e | a vector operator in the spherical basis is | a vector operator in the spherical basis is v = (v, v, v) where the components are: v + 1 = β 1 2 (v x + i v y) v β 1 = 1 2 (v x β i v y), v 0 = v z, {\displaystyle v_{+1}=-{\frac {1}{\sqrt {2}}}(v_{x}+iv_{y})\,\quad v_{-1}={\frac {1}{\sqrt {2}}}(v_{x}-iv_{y})\,,\quad v_{0}=v_{z}\,,} using j Β± = j x Β± i j y, {\textstyl... | wikipedia |
ce38ad2b0cc65129c49bc8871ca7387bfa1675c6 | under rotation of coordinates, the newly defined operator | under rotation of coordinates, the newly defined operator transforms as: u (r) β (v β β
w β) u (r) = u (r) β (β i = 1 3 v i ^ w i ^) u (r) = β i = 1 3 (u (r) β v ^ i u (r)) (u (r) β w ^ i u (r)) = β i = 1 3 (β j = 1 3 r i j v ^ j β
β k = 1 3 r i k w ^ k) {\displaystyle {u(r)}^{\dagger }({\vec {v}}\cdot {\vec {w}})u(r)=... | wikipedia |
1d36712903912a1e87c50f69aef756c03b782f1a | under rotation of coordinates, the newly defined operator | k) v ^ j w ^ k = β k = 1 3 β j = 1 3 Ξ΄ j, k v ^ j w ^ k = β i = 1 3 v ^ i w ^ i {\displaystyle {u(r)}^{\dagger }({\vec {v}}\cdot {\vec {w}})u(r)=\sum _{k=1}^{3}\sum _{j=1}^{3}\left(\sum _{i=1}^{3}r_{ji}^{t}r_{ik}\right){\widehat {v}}_{j}{\widehat {w}}_{k}=\sum _{k=1}^{3}\sum _{j=1}^{3}\delta _{j,k}{\widehat {v}}_{j}{\... | wikipedia |
faf99644e103126c6fa73f141d5aa9a19631c00f | v β β
w β = β i | v β β
w β = β i = 1 3 v i ^ w i ^ {\displaystyle {\vec {v}}\cdot {\vec {w}}=\sum _{i=1}^{3}{\hat {v_{i}}}{\hat {w_{i}}}} | wikipedia |
90c95649c382a892bc03da53f5cf2227664c04c2 | = β c i β Ξ΅ a b | = β c i β Ξ΅ a b c v ^ c {\displaystyle {\left=\sum _{c}i\hbar \varepsilon _{abc}{\widehat {v}}_{c}}} where Ξ΅ is the levi-civita symbol, which all vector operators must satisfy, by construction. the above commutator rule can also be used as an alternative definition for vector operators which can be shown by using the b... | wikipedia |
7acef827273225100fb79c4e1e86f416189698ff | u (r) β v ^ i u (r) | u (r) β v ^ i u (r) = β j r i j v ^ j {\displaystyle {u(r)}^{\dagger }{\widehat {v}}_{i}u(r)=\sum _{j}r_{ij}{\widehat {v}}_{j}} any observable vector quantity of a quantum mechanical system should be invariant of the choice of frame of reference. the transformation of expectation value vector which applies for any wave... | wikipedia |
474585c0e2f594ef55f6f3d9833914342bbfc11d | β¨ Ο | u β (r) a ^ | β¨ Ο | u β (r) a ^ β² u (r) | Ο β© = β¨ Ο | a ^ | Ο β© {\displaystyle \langle \psi |u^{\dagger }(r){\widehat {a}}'u(r)|\psi \rangle =\langle \psi |{\widehat {a}}|\psi \rangle } | wikipedia |
3d26bba5d53e61b6f9ae42f2767d19bd4ee215e0 | | Ο β© β | Ο β² β© | | Ο β© β | Ο β² β© = u (r) | Ο β©, β¨ Ο | β β¨ Ο β² | = β¨ Ο | u β (r) {\displaystyle |\psi \rangle ~\rightarrow ~|\psi '\rangle =u(r)|\psi \rangle \,,\quad \langle \psi |~\rightarrow ~\langle \psi '|=\langle \psi |u^{\dagger }(r)} | wikipedia |
088e780dbb473291d0b4ac9f25bea44e115379f8 | β¨ Ο β² | a β² ^ | | β¨ Ο β² | a β² ^ | Ο β² β© = β¨ Ο | a ^ | Ο β© {\displaystyle \langle \psi '|{\widehat {a'}}|\psi '\rangle =\langle \psi |{\widehat {a}}|\psi \rangle } | wikipedia |
0c546b626b6c9eb743f60154105b4a3481af53f9 | we define the rotation of an operator by | we define the rotation of an operator by requiring that the expectation value of the original operator a ^ {\displaystyle {\widehat {\mathbf {a} }}} with respect to the initial state be equal to the expectation value of the rotated operator with respect to the rotated state, | wikipedia |
23688ba6bebbe70062331fcf0e70648c979824dc | | β, m Β― β© = β m | | β, m Β― β© = β m β² d m β² m (β) | β, m β² β©, | n ^ Β― β© = u (r) | n ^ β© {\displaystyle |{\overline {\ell,m}}\rangle =\sum _{m'}d_{m'm}^{(\ell)}|\ell,m'\rangle \,,\quad |{\overline {\hat {\mathbf {n} }}}\rangle =u(r)|{\hat {\mathbf {n} }}\rangle } | wikipedia |
8ce469791caee49125b181f65ed3e8c60da77976 | so a spherical harmonic can also be written | so a spherical harmonic can also be written y β m = β¨ n | β m β© {\displaystyle y_{\ell }^{m}=\langle \mathbf {n} |\ell m\rangle }. spherical harmonic states | m, β β© {\displaystyle |m,\ell \rangle } rotate according to the inverse rotation matrix u (r β 1) {\displaystyle u(r^{-1})}, while | β, m β© {\displaystyle |\ell,... | wikipedia |
2dab31928e32453c3149b04bd87a802363a6ba14 | n ^ (ΞΈ, Ο) = cos Ο sin | n ^ (ΞΈ, Ο) = cos Ο sin ΞΈ e x + sin Ο sin ΞΈ e y + cos ΞΈ e z {\displaystyle {\hat {\mathbf {n} }}(\theta,\phi)=\cos \phi \sin \theta \mathbf {e} _{x}+\sin \phi \sin \theta \mathbf {e} _{y}+\cos \theta \mathbf {e} _{z}} | wikipedia |
b18fee8a4c3ed75552792f220e92e24a9b3c29e8 | spherical harmonics are functions of the polar and | spherical harmonics are functions of the polar and azimuthal angles, Ο and ΞΈ respectively, which can be conveniently collected into a unit vector n (ΞΈ, Ο) pointing in the direction of those angles, in the cartesian basis it is: | wikipedia |
8007791c0d8bb92b4b24e2d2e26d8c3f6209fc73 | where p is an associated legendre polynomial, β | where p is an associated legendre polynomial, β is the orbital angular momentum quantum number, and m is the orbital magnetic quantum number which takes the values β β, β β + 1,... β β 1, β the formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of sp... | wikipedia |
867194bc8f744aea64e1e3a4fd7122ea62a0c429 | y β m (ΞΈ, Ο) = β¨ ΞΈ, | y β m (ΞΈ, Ο) = β¨ ΞΈ, Ο | β, m β© = (2 β + 1) 4 Ο (β β m) ! (β + m) ! p β m (cos ΞΈ) e i m Ο {\displaystyle y_{\ell }^{m}(\theta,\phi)=\langle \theta,\phi |\ell,m\rangle ={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,p_{\ell }^{m}(\cos {\theta })\,e^{im\phi }} | wikipedia |
2b2fba67f781b6444018bd00e329ed9d28f4417d | for the case of orbital angular momentum, the | for the case of orbital angular momentum, the eigenstates | β, m β© {\displaystyle |\ell,m\rangle } of the orbital angular momentum operator l and solutions of laplace's equation on a 3d sphere are spherical harmonics: | wikipedia |
ecc9031e7adb18d374242ba7e8167df9c2e3f717 | | j, m Β― β© = β m | | j, m Β― β© = β m β² d (r) m β² m (j) | j, m β² β© {\displaystyle |{\overline {j,m}}\rangle =\sum _{m'}{d(r)}_{m'm}^{(j)}|j,m'\rangle } | wikipedia |
cad4fedb571b144a2ff6298466001645c0b6a1fb | | Ο Β― β© = β m m | | Ο Β― β© = β m m β² c j m d m β² m (j) | j, m β² β© β | Ο Β― β© = d (j) | Ο β© {\displaystyle |{\bar {\psi }}\rangle =\sum _{mm'}c_{jm}d_{m'm}^{(j)}|j,m'\rangle \quad \rightarrow \quad |{\bar {\psi }}\rangle =d^{(j)}|\psi \rangle } | wikipedia |
a20897bcd4fe6060216388ec826d875ea012e1a4 | d (r) m β² m (j) = β¨ | d (r) m β² m (j) = β¨ j, m β² | u (r) | j, m β© {\displaystyle {d(r)}_{m'm}^{(j)}=\langle j,m'|u(r)|j,m\rangle } | wikipedia |
69e0c8f287aa4d5ef7ddc43b404184705d94b5eb | | Ο Β― β© = i u (r) | | Ο Β― β© = i u (r) | Ο β© = β m m β² c j m | j, m β² β© β¨ j, m β² | u (r) | j, m β© {\displaystyle |{\bar {\psi }}\rangle =iu(r)|\psi \rangle =\sum _{mm'}c_{jm}|j,m'\rangle \langle j,m'|u(r)|j,m\rangle } | wikipedia |
be1d1d391e1db2e21b311c03c6702d877af14b04 | | Ο Β― β© = u (r) | | | Ο Β― β© = u (r) | Ο β© = β m c j m u (r) | j, m β© {\displaystyle |{\bar {\psi }}\rangle =u(r)|\psi \rangle =\sum _{m}c_{jm}u(r)|j,m\rangle } | wikipedia |
e7880f8b781d3d66f826074cd6590ec2a840bd17 | the orthonormal basis set for total angular momentum | the orthonormal basis set for total angular momentum is | j, m β© {\displaystyle |j,m\rangle }, where j is the total angular momentum quantum number and m is the magnetic angular momentum quantum number, which takes values β j, β j + 1,..., j β 1, j. a general state within the j subspace | wikipedia |
a9fc6ee39728ec6b61808b8643e77b6b65e327b4 | an operator Ξ© ^ {\displaystyle {\widehat {\omega }}} | an operator Ξ© ^ {\displaystyle {\widehat {\omega }}} is invariant under a unitary transformation u if Ξ© ^ = u β Ξ© ^ u; {\displaystyle {\widehat {\omega }}={u}^{\dagger }{\widehat {\omega }}u;} in this case for the rotation u ^ (r) {\displaystyle {\widehat {u}}(r)}, Ξ© ^ = u (r) β Ξ© ^ u (r) = exp (i ΞΈ β n ^ β
j) Ξ© ^ exp ... | wikipedia |
e7e897f95b68dd81c833cb3c14c6277395064fa9 | and let r ^ = r ^ (ΞΈ, | and let r ^ = r ^ (ΞΈ, n ^) {\displaystyle {\widehat {r}}={\widehat {r}}(\theta,{\hat {\mathbf {n} }})} be a rotation matrix. according to the rodrigues' rotation formula, the rotation operator then amounts to u = 1 1 β i sin ΞΈ β n ^ β
j β 1 β cos ΞΈ β 2 (n ^ β
j) 2. {\displaystyle u=1\!\!1-{\frac {i\sin \theta }{\hbar }... | wikipedia |
90b89270aef34d68c9edb74e7441ccccd726edac | j x = β 2 (0 1 0 | j x = β 2 (0 1 0 1 0 1 0 1 0) j y = β 2 (0 i 0 β i 0 i 0 β i 0) j z = β (β 1 0 0 0 0 0 0 0 1) {\displaystyle j_{x}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\,\quad j_{y}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&i&0\\-i&0&i\\0&-i&0\end{pmatrix}}\,\quad j_{z}=\hbar {\begin{pmatrix}-... | wikipedia |
aa96f5f00f0e01511b3adf27d82d4f7a0709bcfc | u = exp (β i ΞΈ β n | u = exp (β i ΞΈ β n ^ β
j) {\displaystyle u=\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {j} \right)} | wikipedia |
0e872e18b5dd8c94a70bcb9cb03f1c0d104625bf | q i j = β Ξ± q Ξ± | q i j = β Ξ± q Ξ± (3 r Ξ±, i r Ξ±, j β r Ξ± 2 Ξ΄ i j). {\displaystyle q_{ij}=\sum _{\alpha }q_{\alpha }\left(3r_{\alpha,i}r_{\alpha,j}-r_{\alpha }^{2}\delta _{ij}\right).} here, the indices i {\displaystyle i} and j {\displaystyle j} can independently take on the values 1, 2, and 3 (or x {\displaystyle x}, y {\displaystyle y... | wikipedia |
ff9e016c70d89705a8ab26c00363e097dd08e51b | scalar, vector and tensor operators can also be | scalar, vector and tensor operators can also be formed by products of operators. for example, the scalar product l β
s {\displaystyle {\mathbf {l} }\cdot {\mathbf {s} }} of the two vector operators, l {\displaystyle {\mathbf {l} }} and s {\displaystyle {\mathbf {s} }}, is a scalar operator, which figures prominently in... | wikipedia |
2c2b8fa0d18ec2841a2f80443688b490f86d5709 | other examples of scalar operators are the total | other examples of scalar operators are the total energy operator (more commonly called the hamiltonian), the potential energy, and the dipole-dipole interaction energy of two atoms. examples of vector operators are the momentum, the position, the orbital angular momentum, l {\displaystyle {\mathbf {l} }}, and the spin ... | wikipedia |
df0ac9d463a1e35072c89fcde002cbdc6c164f06 | in the same way, tensor quantities must be | in the same way, tensor quantities must be represented by tensor operators. an example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively. | wikipedia |
2b9d77a1d7dc99026bfc43d93f603841a6a35325 | in quantum mechanics, physical observables that are scalars, | in quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. alt... | wikipedia |
dbda23121f319bdae077a8e9a013a65d29b2a18c | in pure and applied mathematics, quantum mechanics and | in pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. a special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. the spherical basis closely relate... | wikipedia |
32c0a8f4274a2b30607a83c068ae5b31111906d0 | trevor lewis is an american professional ice hockey | trevor lewis is an american professional ice hockey center for the los angeles kings of the national hockey league (nhl). lewis appeared in parts of 12 seasons with the kings after being drafted 16th overall by the team in the 2006 nhl entry draft; he spent one season with the winnipeg jets before signing with the calg... | wikipedia |
b2d99bee6b9114e32c700a1f770e1f5407775c27 | on july 28, 2021, lewis signed a one-year, | on july 28, 2021, lewis signed a one-year, $800,000 contract with the calgary flames, reuniting him with former kings head coach darryl sutter. lewis earned his first point with the flames, an assist in a game against the new york rangers. for only the second time in his career, lewis played a full 82-game season in 20... | wikipedia |
66e7b403f872e1d11165aa6289a7edf21f47820f | as a free agent leaving the kings organization | as a free agent leaving the kings organization after 12 seasons, lewis remained unsigned leading into the pandemic-delayed 2020β21 season. he accepted an invitation to join the winnipeg jets training camp on a professional tryout basis and upon impressing was later signed to a one-year, $750,000 contract by the jets on... | wikipedia |
9b2d97115759f1c786f633c42c98e4245119b704 | during the following 2017β18 season, lewis put up | during the following 2017β18 season, lewis put up a career-high 26 points despite being placed on injured reserve in february. after appearing in 17 games for the kings during the 2018β19 season, and recording three points, lewis was again placed on injured reserve due to a lower-body injury. he was activated off injur... | wikipedia |
42acbceea4a5b99b79abc690aab1e6de279c2b87 | he signed a four-year contract with the kings | he signed a four-year contract with the kings on june 25, 2016. it would pay off, as in the 2016β17 season, lewis would score an equal 12 goals and assists for 24 points, playing a full 82 games for the first time in his career. | wikipedia |
a6b9e6ade20554fd81efff06212a67b0b15f50b5 | in the 2011β12 season, on june 11, 2012, | in the 2011β12 season, on june 11, 2012, lewis won the stanley cup as a member of the los angeles kings, their first championship in franchise history. he scored two goals in the clinching game six. lewis signed another two-year extension with the kings on april 8, 2014, before helping the kings to their second stanley... | wikipedia |
4e01b2460c72a509d0dbc165ee06d97a5b91d8ca | lewis was drafted 17th overall by the los | lewis was drafted 17th overall by the los angeles kings in the 2006 nhl entry draft, following an award-winning season in the united states hockey league with the des moines buccaneers. on july 14, 2006, the kings signed lewis to a three-year entry-level contract. he played the 2006β07 season with the owen sound attack... | wikipedia |
27e0d6f54ef5f2789d3324b51931254025c14603 | the son of a transplanted canadian, lewis grew | the son of a transplanted canadian, lewis grew up in salt lake city where he learned to skate at the age of two. he began playing hockey at the age of five, eventually moving to colorado springs, colorado at the age of 14 to play for the pike's peak miners aaa team. | wikipedia |
97d7409a4734321fcbee21804a3e0de882afa0a1 | trevor lewis (born january 8, 1987) is an | trevor lewis (born january 8, 1987) is an american professional ice hockey center for the los angeles kings of the national hockey league (nhl). lewis appeared in parts of 12 seasons with the kings after being drafted 16th overall by the team in the 2006 nhl entry draft; he spent one season with the winnipeg jets befor... | wikipedia |
5b3f6aeca54d6fcb10fdc8fbd6f99d8de0a36c31 | lms, is a color space which represents the | lms, is a color space which represents the response of the three types of cones of the human eye, named for their responsivity (sensitivity) peaks at long, medium, and short wavelengths. the numerical range is generally not specified, except that the lower end is generally bounded by zero. it is common to use the lms c... | wikipedia |
6b6542d40ebfa038864ef4224da228bcbb4d971b | this can be interpreted as a hybrid color | this can be interpreted as a hybrid color theory where l and m are opponents but s is handled in a trichromatic way, justified by the lower spatial density of s cones. in practical terms, this allows for using less data for storing blue signals without losing much perceived quality. | wikipedia |
dba56f9f342624b2b1e28b993c9dd13361aa3d18 | jpeg xl uses an xyb color space derived | jpeg xl uses an xyb color space derived from lms. its transform matrix is shown here: = {\displaystyle {\begin{bmatrix}x\\y\\b\end{bmatrix}}={\begin{bmatrix}1&-1&{\phantom {-}}0\\1&{\phantom {-}}1&{\phantom {-}}0\\0&{\phantom {-}}0&{\phantom {-}}1\end{bmatrix}}{\begin{bmatrix}l\\m\\s\end{bmatrix}}} | wikipedia |
381ef548989b8142e3b36f0115191c8430073654 | the lms color space can be used to | the lms color space can be used to emulate the way color-blind people see color. an early emulation of dichromats were produced by brettel et al. 1997 and was rated favorably by actual patients. an example of a state-of-the-art method is machado et al. 2009. | wikipedia |
65feef8800b08247601ce719a7104bed5a417abf | if ce (Ξ») (i =1,2,3) are the three | if ce (Ξ») (i =1,2,3) are the three energy-based color matching functions for a particular color space (lms color space for the purposes of this article), then the tristimulus values may be expressed in terms of the quantal radiative quantity by: | wikipedia |
030a5da41d2ebc4c1bf6d692249bab79ddb8ab4c | where e is the energy per photon, h | where e is the energy per photon, h is the planck constant, c is the speed of light, Ξ½ is the frequency of the radiation and Ξ» is the wavelength. a spectral radiative quantity in terms of energy, je (Ξ»), is converted to its quantal form jq (Ξ») by dividing by the energy per photon: | wikipedia |
a8814c482296b989bbce9585efc9770c7cf2561c | the above development has the advantage of basing | the above development has the advantage of basing the new x y z color matching functions on the physiologically-based lms cone response functions. in addition, it offers a one-to-one relationship between the lms chromaticity coordinates and the new x y z chromaticity coordinates, which was not the case for the cie 1931... | wikipedia |
9abb4c96ef993fc4cec666b3b02fb8d1bf6ad074 | the inverse matrix is shown here for comparison | the inverse matrix is shown here for comparison with the ones for traditional xyz: = f {\displaystyle {\begin{bmatrix}l\\m\\s\end{bmatrix}}=\left{\begin{bmatrix}x\\y\\z\end{bmatrix}}_{\text{f}}} | wikipedia |
35680a62217c89df25635d7a2fcf73bf8c5b135c | or, explicitly: f = {\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}_{\text{f}}=\left{\begin{bmatrix}l\\m\\s\end{bma | or, explicitly: f = {\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}_{\text{f}}=\left{\begin{bmatrix}l\\m\\s\end{bmatrix}}} | wikipedia |
8615237a15e1a0642c6239ca3ad380759d1dbfef | for any spectral distribution j (Ξ») {\displaystyle j(\lambda)}, | for any spectral distribution j (Ξ») {\displaystyle j(\lambda)}, let p i = (l, m, s) {\displaystyle p_{i}=(l,m,s)} be the lms chromaticity coordinates for j (Ξ») {\displaystyle j(\lambda)}, and let q i = (x, y, z) f {\displaystyle q_{i}=(x,y,z)_{\text{f}}} be the corresponding new xyz chromaticity coordinates. then: | wikipedia |
a9960542629a0afeee0a1502bbaec9532d6f2204 | let p i (Ξ») = (l Β― (Ξ»), | let p i (Ξ») = (l Β― (Ξ»), m Β― (Ξ»), s Β― (Ξ»)) {\displaystyle {\mathcal {p}}_{i}(\lambda)=({\bar {l}}(\lambda),{\bar {m}}(\lambda),{\bar {s}}(\lambda))} be the three cone response functions, and let q i (Ξ») = (x Β― f (Ξ»), y Β― f (Ξ»), z Β― f (Ξ»)) {\displaystyle {\mathcal {q}}_{i}(\lambda)=({\bar {x}}_{\text{f}}(\lambda),{\bar {... | wikipedia |
6a0dec0dbe96cd17c749a3195df0bf165cfb7ac1 | a set of physiologically-based lms functions were proposed | a set of physiologically-based lms functions were proposed by stockman & sharpe in 2000. the functions have been published in a technical report by the cie in 2006 (cie 170). the functions are derived from stiles and burch rgb cmf data, combined with newer measurements about the contribution of each cone in the rgb fun... | wikipedia |
92332b1b1cf6849e71fa21abd474d2204dc5ebec | cam16 uses a different matrix: 16 = {\displaystyle | cam16 uses a different matrix: 16 = {\displaystyle {\begin{bmatrix}r\\g\\b\end{bmatrix}}_{\text{16}}=\left{\begin{bmatrix}x\\y\\z\end{bmatrix}}} | wikipedia |
42c2bf9da962f3b578e93036cf75e18803e2da0c | the sharpened transformation matrix in ciecam02 (m) is: | the sharpened transformation matrix in ciecam02 (m) is: 02 = {\displaystyle {\begin{bmatrix}r\\g\\b\end{bmatrix}}_{\text{02}}=\left{\begin{bmatrix}x\\y\\z\end{bmatrix}}} | wikipedia |
09dbad6fb9acd2d72d558c6e6b356eea323cbee4 | a revised version of ciecam97s switches back to | a revised version of ciecam97s switches back to a linear transform method and introduces a corresponding transformation matrix (m): 97 = {\displaystyle {\begin{bmatrix}r\\g\\b\end{bmatrix}}_{\text{97}}=\left{\begin{bmatrix}x\\y\\z\end{bmatrix}}} | wikipedia |
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