Equal commutators

  • #1
302
0
Hi.

Cohen-Tannoudji has this section in his quantum mechanics book where he derives a bunch of relations which are true for operators having the commutation relation [itex][Q,P]=i\hbar[/itex]. Is there any special significance to this value of a commutator? Would things be much different if it had the value 1 ?

Also, if we have two sets of operators with the same commutator, i.e. [itex][x,p]=[Q,P]=i\hbar[/itex], what does this tell us about the relations between the operators, if anything?
 

Answers and Replies

  • #2
nicksauce
Science Advisor
Homework Helper
1,272
5
Take the hermitian conjugate of each side. If the operators are Hermitian, then you get
[P,Q] = -ih = -[Q,P], which is exactly what you expect based on the definition of a commutator. If you had [Q,P] = 1, then this process would lead to a contradiction if your operators are Hermitian.
 
  • #3
302
0
Ah, so that's why it's important that it's imaginary! Great. What about sets of operators with the same commutator? I know that if we have two Hamiltonians of the same form, i.e.
[tex]
H=a^{\dagger} a+\frac{1}{2}
[/tex]
[tex]
H=b^{\dagger} b+\frac{1}{2}
[/tex]

and [itex][a^{\dagger},a]=[b^{\dagger},b][/itex] then the Hamiltonians will have the same eigenvalues. Is there more we can say? I've heard that all of quantum mechanics can be based on commutators...
 
  • #4
nicksauce
Science Advisor
Homework Helper
1,272
5
I'm not too sure what we can say about sets of operators with the same commutator... hopefully someone else can help you out.
 

Related Threads on Equal commutators

  • Last Post
Replies
10
Views
8K
Replies
3
Views
789
  • Last Post
Replies
9
Views
3K
  • Last Post
Replies
11
Views
846
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
6
Views
840
  • Last Post
Replies
4
Views
1K
  • Last Post
2
Replies
38
Views
6K
  • Last Post
Replies
3
Views
662
Top