• Support PF! Buy your school textbooks, materials and every day products Here!

Question about a Problem from Sakurai

  • Thread starter Xyius
  • Start date
  • #1
508
4
Hello!

I am studying the Sakurai book on Quantum mechanics and I am doing a problem. I have the solutions to the problems to help me understand the material better but I do not understand this solution.

Homework Statement



SEE "Sakurai Problem 1" in attachments

K is the propagator in wave mechanics.


2. Solution

SEE "Sakurai Problem 1" in attachments

There are a few parts of this solution that I do not understand.

1.) In the first part it states that "The probability is.."

[tex]P(Ea')=exp(-βEa')/Z[/tex]

Probability of what? It doesn't actually tell me what the "Partition function" is or means. Isn't the propagator an operator? I thought in order to have a probability you need to have a state in mind.

2.) I also do not understand why the ground state energy is equal to that summation "U=...." in the next line.

3.) I do not understand the first change of variables in the differential, da' = L/2π dk

If anyone could help me understand this, it would be much appreciated! :D
 

Attachments

Answers and Replies

  • #2
DrDu
Science Advisor
6,032
759
If beta were an inverse temperature, P would be the probability to find E_a' in a canonical ensemble.
The precise probability interpretation does not matter too much, as all you want to show is that the limit on beta gives you the ground state energy.
No, the propagator isn't an operator but a matrix element of the the time evolution operator between states <x',t'| and |x,0>.
 
  • #3
1,024
32
If you will see the text,you will find that space integral of propagator with x''=x' in K(x'',t;x',t0) will give you G(t)=Ʃa'exp(-iEa't/h-).This is just the trace of time evolution operator and is independent of representation.Now you have to identify β=it/h-(with t imaginary),and you will identify it as partition function.
 
  • #4
508
4
Thank you for the replies.

If beta were an inverse temperature, P would be the probability to find E_a' in a canonical ensemble.
I still do not seem to understand. How do they actually obtain the expression for probability? I don't understand the reasoning behind it.

andrien said:
If you will see the text,you will find that space integral of propagator with x''=x' in K(x'',t;x',t0) will give you G(t)=Ʃa'exp(-iEa't/h-).This is just the trace of time evolution operator and is independent of representation.Now you have to identify β=it/h-(with t imaginary),and you will identify it as partition function.
I didn't even know what a partition function was until I just looked it up. I never took statistical mechanics. :\
 

Related Threads on Question about a Problem from Sakurai

  • Last Post
Replies
1
Views
3K
  • Last Post
2
Replies
28
Views
8K
  • Last Post
Replies
3
Views
4K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
5
Views
4K
  • Last Post
Replies
3
Views
6K
Replies
2
Views
858
Replies
7
Views
1K
  • Last Post
Replies
7
Views
4K
Replies
1
Views
2K
Top