Question about a Problem from Sakurai

[Xyius]

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

 

[DrDu]

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>.

 

[andrien]

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)=Ʃ_a'exp(-iE_a'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.

 

[Xyius]

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.

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. :\

 

[paranormal]

Hello!

I would be happy to help clarify some of the points you are struggling with in this problem.

1. The probability in this case refers to the probability of the system being in a particular energy state, Ea'. The partition function, Z, is used to normalize the probability distribution, and it is related to the sum of all possible states of the system. So, in order to calculate the probability of being in the state Ea', we need to know the partition function. The propagator, K, is indeed an operator, but it is used to calculate the probability of transitioning from one state to another. In this case, it is used to calculate the probability of the system being in the state Ea' at a certain time.

2. The ground state energy, U, is equal to the summation because it is the sum of all possible energies of the system. In this case, the system can only be in discrete energy states, so the ground state energy is the lowest possible energy state, which is the sum of all the individual energies.

3. The change of variables in the differential, da' = L/2π dk, is a common technique used in solving integrals. It is a way to change the variable of integration from a' to k. The L/2π factor is there to account for the different units of a' and k.

I hope this helps clarify some of the confusion you had with this problem. Don't hesitate to ask for further clarification if needed. Keep up the good work studying Quantum mechanics!