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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.

SEE "Sakurai Problem 1" in attachments

K is the propagator in wave mechanics.

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

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