Five State Quantum System, understanding the question

[Cetus]

Homework Statement

A simple system has different quantum states with energies -1, 0, 0, 1, 1 (x10^-20 Joules)
Determine the probability that the system is in these different states, if T= 400K.

Relevant Equations

$$E_{thermal} = k_b T$$

I’ve never worked with a quantum system with more that two states 1, -1, and I’ve just gotten this homework problem. I'm not sure what it means. Does this mean it has five states? Why are there two 0’s and two 1’s?

 

[PeroK]

The question tells you the system has five states. There's no law that says that two different states can't have the same energy. E.g. different states of the hydrogen atom have the same total energy (but differ in other respects). In general, this is called "degeneracy".

Apart from that, I must admit I don't understand the question. I guess that the given $E$ is the expected value of the energy of the system. So, it must be in a superposition of energy eigenstates.

Note that the question asks for "probability that the system is in these different states". If the system is in a superposition, then it is not in any of these states. That's at best sloppy language and at worst just plain wrong.

There is a probability that, on measuring the energy of the system, you will get $-1, 0, 1$, and then you can say that, after measurement, it has a definite energy. Even then, because of degeneracy, you would only know it is in a superposition of the appropriate energy eigenstates.

Is this the whole question?

PS

 

[kith]

I’ve never worked with a quantum system with more that two states 1, -1, and I’ve just gotten this homework problem.

How would you solve the problem if there were only the two states you mention?

 

[PeroK]

PPS Something else I don't understand about this question:

If you get the $-1$ eigenstate, then that would imply a negative temperature. I don't see how, with thermal energy defined as $E = k_b T$, we can have a negative energy eigenstate.

 

[kith]

Apart from that, I must admit I don't understand the question. I guess that the given $E$ is the expected value of the energy of the system. So, it must be in a superposition of energy eigenstates.

The energy of the system isn't given, its temperature is. So we have a thermal mixed state where the question of probabilities is more sensible.

As far as the temperature is concerned: $k_b T$ is something like the mean energy with which every degree of freedom of the system at temperature $T$ is excited. That a certain energy level of the system is negative isn't related to the temperature in a specific physical situation (the ground state energy of the hydrogen atom, for example, is also negative).

 

[PeroK]

(the ground state energy of the hydrogen atom, for example, is also negative).

Yes, but the energy in that case is defined in terms of a Coulomb potential. Anyway, it's not my homework. I'm just saying that if it were my homework, I wouldn't know what to make of it!

 

[Dr_Nate]

It is a poorly worded question. I'm guessing since it doesn't specify boson or fermion that this system has one particle. I think that they are expecting a thermodynamic partition function approach.

 

[DrClaude]

I don't think that the question is badly posed, only that some context is missing. Although, as I am teaching Stat. Phys. at the moment, the context appears obvious to me

Relevant Equations:: $$E_{thermal} = k_b T$$

This doesn't make sense.

When faced with such problems, the first thing you should do is to calculate the partition function.