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Background

Consider the measurement. The measurement maps an arbitrary energy eigenstate $|E_i \rangle$ to $| \phi \rangle$:

$$|E_i \rangle \to | \phi \rangle$$

with a probability of this outcome is $|\langle E_i | \phi \rangle|^2$. Note the density matrix remains unaltered.

What if I have $2$ Hamiltonians $H$ and $\tilde H$ with eigen energies $|E_i \rangle$ and $|\tilde E_j \rangle$. Let us assume they are of the same inverse temperature $\beta$. I place them in contact and by zero'th law of the probability of finding a particular energy or temperature should remain the same.

Thus, a possible mapping seems to be

$$ |E_i \rangle \otimes |\tilde E_j \rangle \to | \phi_{\beta} \rangle \otimes |\lambda_\beta \rangle$$

where the probability of this outcome is $?$

One can think of the measurement as a possible mapping where the average energy(/momentum/any other symmetry) here is conserved. With that in mind this mapping can be employed to conserve probability (take into account the probability of finding $E_i$ and $\tilde E_j$ is $\exp(-\beta (E_i + \tilde E_j)$) is but not energy?

We know by the sudden approximation if I "suddenly" put them in contact:

$$ H \otimes 1 + 1 \otimes \tilde H \to H \otimes 1 + 1 \otimes \tilde H + H_{contact} $$

then the eigenstates won't have time to react and thus remain the same. One (false) way explaining away the situation is to say $H_{contact}$ is negligible and one can employ the sudden approximation. However, if that were the case then we should be able to neglect $H_{contact}$ even when $H$ and $\tilde H$ are of different temperatures!

Question

How does the literature resolve this contradiction? (Feel free to confine this study to a specific physical system as well)

More Anonymous
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  • If "literature resolves this contradiction" it would do so by using an accurate model for the physical system involved. Note that energy is a property that effectively exists for all time. The idea of putting two subsystem "suddenly" in contact with each other (whatever that means) is something that happens in time. Therefore the energy of the combined system cannot be the same as (or the sum of) the individual systems. Unless, you want to treat the contact as a perturbation, you would need to solve the combined system as a whole. – flippiefanus Jun 03 '22 at 11:06
  • @flippiefanus I get that my question is: I get that why is that when it is of the same temperature one can use the sudden approximation and then the probability remains the same. On the other hand other hand when they are of different temperatures apparently the sudden approximation? – More Anonymous Jun 03 '22 at 11:49
  • Hi More Anonymous, asking about literature makes this post a res. recom q, which is tightly regulated on Phys.SE. To reopen this post either make it a pure res. recom q, or remove explicit literature requests. – Qmechanic Jun 03 '22 at 15:53

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