# What is the relation between Temperature and Quantum State?

• I
Hi folks,

Let's pick a simple example, the H atom. We can calculate all spherical armonics, all quantum numbers so we are able to know which are all the possible states of the electron. We know all the values this observables can take. But the question is, let's say we have a handbook of properties from Hydrogen and we find that at temperature T, the electrical conductivity is X. What is the corresponding state of the atom at that temperature?

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vanhees71
Gold Member
I don't understand what you are after. Temperature makes sense in thermal equilibrium only, and this is described by the grand canonical ensemble, i.e., the state
$$\hat{\rho}=\frac{1}{Z} \exp(-\beta \hat{H}), \quad Z=\mathrm{Tr} \exp(-\beta \hat{H}),$$
where ##\hat{H}## is the Hamiltonian of the (many-body) system. That's the relation between temperature ##T=1/\beta## (in natural units, where ##k_{\text{B}}=1##) and the state, reprsented by the grand-canonical statistical operator.

jonjacson
I don't understand what you are after. Temperature makes sense in thermal equilibrium only, and this is described by the grand canonical ensemble, i.e., the state
$$\hat{\rho}=\frac{1}{Z} \exp(-\beta \hat{H}), \quad Z=\mathrm{Tr} \exp(-\beta \hat{H}),$$
where ##\hat{H}## is the Hamiltonian of the (many-body) system. That's the relation between temperature ##T=1/\beta## (in natural units, where ##k_{\text{B}}=1##) and the state, reprsented by the grand-canonical statistical operator.

I have found a related thread, I apologize for that:

Well what I wanted to understand is Superconductivity. We have a solid, and we know at certain temperature it goes to superconductor mode and energy loss for the current is 0. But I wanted to understand what it means from the quantum point of view. Since temperature is not a quantum magnitude I wanted to know how it relates to quantum mechanics.

In superconductivity you have a solid, you have its behavior under different temperatures regarding its electric conductivity. Do you know if there is any theory that tells you at what temperature the solid goes to superconductivity mode?

I mean, if I give you as an input a unit Bravais cell, Would you be able to tell me the temperature for superconductivity?

vanhees71
Gold Member
Look for "BCS theory" which describes superconductivity quantum(-field) theoretically. A very good book about superfluidity and superconductivity is

A. Schmitt, Indroduction to Superfluidity, Springer (2015)

jonjacson
Look for "BCS theory" which describes superconductivity quantum(-field) theoretically. A very good book about superfluidity and superconductivity is

A. Schmitt, Indroduction to Superfluidity, Springer (2015)

THanks, I will do.