text stringlengths 0 8.13M |
|---|
equation. The states does not even appear here. But in fact, there is a close |
relation and it will be explained in chapter 5. Suffice it here to say the present |
form of dynamics, which goes under the name Heisenberg picture, the time- |
development of the system is carried by the operators while the states are |
time-independent. That is the reason why they don’t appear explicitly. In |
the Schr¨odinger equation approach, called the Schro¨dinger picture, the states |
carry the time-development while the operators are time-independent. |
4.2.6 Dirac notation, a case of abstraction |
The Dirac notation is a very useful way of formulating quantum mechanics |
in that it allows quantum mechanical systems to be treated in a uniform way |
by abstracting away from particularities. In fact, a computer scientist might |
want to regard the Dirac formalism as providing an interface, specifying what |
properties and methods a system should support without entering into details |
on how to implement them. |
Suppose a certain system is described by a wave function ψ(x) expanded as |
in (5.13) |
∞ |
ψ(x)= α u (x). |
n n |
nX=0 |
Now,thebasisfunctionsu belongtoacertainclassoffunctions,allsharing |
n |
commonproperties. Oftenitisjustthosepropertiesthatareimportant,notthe |
explicit x-space representation. Letting n label these properties, we can write |
the expansion in the form |
∞ |
ψ = α n , |
n |
| i | i |
nX=0 |
where the ket-notation wasintroducedby Diracto denote abstractquan- |
|·i |
tum states. |
Instead of having physical quantities represented as explicit differential op- |
erators acting on the wave functions, those quantities are now represented as |
abstract operators acting on the label n. |
81 |
4.2.7 Summary of the classical harmonic oscillator |
Returningnowtotheharmonicoscillator,wecanwritethedynamicalequations |
compactly using the general formalism just developed. The Hamiltonian is |
p2 kx2 |
H = + . |
2m 2 |
WiththisparticularformfortheHamiltonian,equations(4.39)and(4.40)yield |
dx p2 kx2 p2 p |
= x, + = x, = |
dt { 2m 2 } { 2m} m |
dp p2 kx2 kx2 |
= p, + = p, = kx. |
dt { 2m 2 } { 2 } − |
Together these equations give |
d2x |
m = kx |
dt2 − |
which is Newton’s equation for a harmonic oscillator. To see this, note that |
the first equation gives p = mv and insert this formula for p in the second |
equation. |
4.2.8 Creation and annihilation operators |
The groundwork is now laid for treating the harmonic oscillator using creation |
and annihilation operators. First, the classical Hamiltonian is rewritten in a |
suggestive way |
1 p2 1 k p2 |
H = +kx2 = +√kmx2 . |
2 m 2rm √km |
(cid:0) (cid:1) (cid:0) (cid:1) |
Introducing ω = k/m, the Hamiltonian H becomes7. |
p |
1 p2 |
H = ω +mωx2 . (4.43) |
2 mω |
(cid:0) (cid:1) |
Inanticipationofquantummechanics,introduce¯handdoafurtherrewriting |
1 p2 mω |
H = ¯hω + x2 = |
2 mω¯h ¯h |
(cid:0) (cid:1) |
1 p mω p mω |
¯hω i x +i x = |
2 √mω¯h − r ¯h √mω¯h r ¯h |
(cid:0) (cid:1)(cid:0) (cid:1) |
1 |
¯hωzz¯, |
2 |
or for short |
7ωisrelatedtotheclassicalfrequencyofoscillationsf throughω=2πf. Ausefulformula |
ismω=√km |
82 |
1 |
H = ¯hωzz¯. (4.44) |
2 |
TheintuitionbehindthisrewritingisthatH isaquadraticform,andthere- |
fore it should be possible to write it as a square. However, since x and p will |
become operators, care must be exercised regarding the order in which they |
appear in the product zz¯. From now on, x and p are treated as operators, but |
we will drop the ’hat’ notation, writing A for Aˆ. |
InordertocapitalizeonthepossibilitytowriteH asasquare,definecreation |
and annihilation operators a and a |
† |
1 |
a= (p imωx) (4.45) |
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