Chapter 9 Lecture 9 - Compatibility of Observables and Communtation Relations
9.1 Compatible and Incompatible Observables
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Compatible Observables:
- some observables can be sumultaneously measured. Determining more information about one does not affect the information we have about the other. For example x and y , x and x2 and px and py
- Incompatible Observables:
- for some observables however determining more inromation about one reduces the information we can simultaneously have about the other. For example x and px also x2 and px
The spread in the value of an observable D A ( A operator) is a measure of this information. If D A is large then only a little information is lost, however if D A is small then a lot of information is lost (the uncertainity increases).
The spread is more formally defined as a standard deviation.
D A=<A2>-<A>2 (9.1)
If a pair of observables A and B are compatible then there must exist a set of eigenstates ( Yn ) common to both
AYn=anYn BYn=bnYn (9.2)
If a pair of observables A and B are incompatible then we cannot find a set of eigenstates ( Yn ) common to both.
9.2 Commutators
We define a commutator of two operators in the following way
[A, B]=AB-BA (9.3)
it follows that
[A, B]=-[B, A] (9.4)
For compatible observables Yn is an eigenstate of both
[A, B]Yn=(AB-BA)Yn=ABYn-BAYn
note that ABYn means B operates first on Yn then A operates on the result.
[A, B]Yn=AbnYn-BanYn=anbnYn-bnanYn=0
this means if the following is true then we have compatible operators
[A, B]=0 (9.5)
This is an important test of compatibility.
For incompatible observables
[A, B]Y =ABY-BAY
Y is not an eigenstate of both
ABY¹BAY
So our test for incompatible observables is
[A, B]¹ 0 (9.6)
If [A, B]¹ 0 then there exists an uncertainity relation D AD B .
9.3 Some Important Communtation Relations
9.3.1 Example 1
[x, px]
we recall that
x=x
always remember that operators should operate on a function, so we write
| [x, px]Y(x)=- |
é
ê
ê
ë |
x |
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- |
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ù
ú
ú
û |
Y(x)=- |
é
ê
ê
ë |
x |
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Y(x)-Y(x) |
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-x |
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Y(x) |
ù
ú
ú
û |
=Y(x) |
[x, px]= (9.7)
therefore we have uncertainity in the simultaneous measurement of x and px .
9.3.2 Example 2
[H, px]
so in one dimension
so we can say this comutes with px
[H, px]=[V(x), px]
| [V(x), px]=- |
é
ê
ê
ë |
V(x) |
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- |
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V(x)-V(x) |
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ù
ú
ú
û |
= |
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V(x) |