Abstract
Consider a quantum system S weakly interacting with a very large but finite system B called the heat bath, and suppose that the composite S∪B is in a pure state Ψ with participating energies between E and E+δ with small δ. Then, it is known that for most Ψ the reduced density matrix of S is (approximately) equal to the canonical density matrix. That is, the reduced density matrix is universal in the sense that it depends only on S’s Hamiltonian and the temperature but not on B’s Hamiltonian, on the interaction Hamiltonian, or on the details of Ψ. It has also been pointed out that S can also be attributed a random wave function ψ whose probability distribution is universal in the same sense. This distribution is known as the “Scrooge measure” or “Gaussian adjusted projected (GAP) measure”; we regard it as the thermal equilibrium distribution of wave functions. The relevant concept of the wave function of a subsystem is known as the “conditional wave function.” In this paper, we develop analogous considerations for particles with spin. One can either use some kind of conditional wave function or, more naturally, the “conditional density matrix,” which is in general different from the reduced density matrix. We ask what the thermal equilibrium distribution of the conditional density matrix is, and find the answer that for most Ψ the conditional density matrix is (approximately) deterministic, in fact (approximately) equal to the canonical density matrix.
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Notes
By a GONB we mean that which is provided by a unitary isomorphism
for some measure space 
containing the y; this includes the possibility of a continuous basis such as the position basis.This factor, \(\mathcal{N}=\|\varPsi(\cdot ,Y)\|\), will fail to be well defined if Ψ(⋅,Y) fails to be square-integrable. However, the Y for which this happens form a set of measure zero because ∫dy ∥Ψ(⋅,y)∥2=∫dy∫dx |Ψ(x,y)|2<∞. The factor \(\mathcal{N}\) could be zero, but since Y has distribution density ∥Ψ(⋅,y)∥2, also this case occurs with probability zero.
If
, we can also admit a GONB. Since we assume here that 
, every GONB is an ONB.The point here is that we can change the order of the quantifiers (“for most y” etc.) without changing the content of the statement; this was not possible as long as the notion of “most y” depended on Ψ.
Actually, we use here more than just large
; we use that among the eigenvalues of \(\rho_{\beta}^{S}\otimes\rho_{\beta}^{s}\) are not just a few dominating ones while all others are negligible, but that many of them are of comparable size; that is the case since H
s
has a reasonable distribution of eigenvalues.
for some measure space 
containing the y; this includes the possibility of a continuous basis such as the position basis.
, we can also admit a GONB. Since we assume here that 
, every GONB is an ONB.
; we use that among the eigenvalues of References
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R.T. was supported by grant no. 37433 from the John Templeton Foundation.
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Pandya, V., Tumulka, R. Spin and the Thermal Equilibrium Distribution of Wave Functions. J Stat Phys 154, 491–502 (2014). https://doi.org/10.1007/s10955-013-0849-y
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DOI: https://doi.org/10.1007/s10955-013-0849-y