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I'm reading in Feynman's "Statistical Mechanics" Chap. 6.4 about a system of $M$ interacting particles, they may be bosons or fermions. Let the hamiltonian be

$$ H=\sum_i^{3M}p_i^2+\sum_{ij}^{3M}U_{ij}q_iq_j,\tag{1} $$

with a symmetric matrix $U$ and apropriate definition of p and q to get rid of the constants. By diagonalizing, the Hamiltonian can be brought in the form $$H=\frac{1}{2}\sum_i^{3M}(P_i^2+\omega_i^2Q_i^2),\tag{2}$$ with generalized coordinates $P_i$ and $Q_i$. Define the creation and anihilation operators $a_k=\frac{1}{\sqrt{2\hbar}}(\sqrt\omega_kQ_k+\frac{i} {\sqrt{\omega_k}}P_k)$ and $a_k^\dagger$.

The hamiltonian therefore takes the form $$ H =\sum_i^{3M}\hbar\omega_i\left(a_i^\dagger a_i+\frac{1}{2}\right) =\sum_i^{3M}\hbar\omega_i\left(N_i+\frac{1}{2}\right).\tag{3} $$

The eigenstates are $$ |n_1n_2...n_{3M}\rangle =\prod_i^{3M}\frac{(a_i^\dagger)^{n_i}}{\sqrt{n_i!}}|\mathrm{vac}\rangle \tag{4} $$ $$ H|n_1n_2...n_{3M}\rangle =\sum_i^{3M}\hbar\omega\left(n_i+\frac{1}{2}\right) |n_1n_2...n_{3M}\rangle, \tag{5} $$

and they are interpreted as $n_1$ "phonons" in the first mode, $n_2$ in the second and so on.

Now to the questions:

  1. How do we have to apply the Pauli principle here? I suppose the states are tensor products of "single particle" states $|n_1n_2...n_{3M}\rangle:=|n_1\rangle|n_2\rangle...|n_{3M}\rangle$, but in that case they would not be symmetrized properly, since for example $a_1^\dagger(a_2^\dagger)^3 |\mathrm{vac}\rangle=|1\rangle|3\rangle $ is neither a symmetric nor an antisymmetric state.

  2. How do we even know what the proper symmetry condition ought to be? Does the wavefunction have to be symmetric under exchange of the phonon mode?

  3. Does (2.) depend on whether the interacting particles from the start are bosons or fermions, or rather on the type of the mode? What if it is a mix of different kinds or particles?

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The Pauli principle is already in full effect - it is enforced at the time that you set your creation and annihilation operators to obey canonical commutation relations of the form $$ [a_i^\phantom{\dagger},a_j^\dagger]=\delta_{ij}, $$ as opposed to anti-commutation relations of the form $\{a_i^\phantom{\dagger}, a_j^\dagger\} =\delta_{ij}$. For the phonon case you're dealing with the canonical commutation relations are a consequence of the underlying structure rather than an externally-imposed axiom, but the result is the same, as it is the (anti)commutation relations that mark the distinction between fermions and bosons on any second-quantized formalism like the one you're using.

Among other things, this means that your claim that

for example $a_1^\dagger(a_2^\dagger)^3 |\mathrm{vac}\rangle=|1\rangle|3\rangle $ is neither a symmetric nor an antisymmetric state

isn't correct - the state $|1\rangle|3\rangle$ already is fully symmetrized. What matters isn't the wavefunction's symmetry under, in your words,

exchange of the phonon mode,

which doesn't make sense - what you'd care about is exchange of the phonons within each mode or between different modes: i.e. a symmetry operation that takes one of those three photons inside that $|3⟩$ and exchanges it for the second one of those. Or a symmetry operation that takes the photon in the $|1⟩$, puts it in the $|3⟩$, and then takes one of the original $|3⟩$ and puts that inside the same mode that started in $|1⟩$.

When phrased like that, of course, those symmetry operations don't even make any sense at all, and that is because you're already operating on an automatic second-quantized formalism which, regardless of where it came from, renders such questions completely moot. The exchange symmetry is encoded in the (anti)commutation relations and that's it.

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  • $\begingroup$ I think I`m getting it, but what still confuses me is this: I can ask for the probability that the first generalized coordinate takes the value $Q_1$ and the second the value $Q_2$, lets say for the state that I wrote in the question: $P(Q_1,Q_2)=|\langle Q_1|\langle Q_2|)(|1\rangle|3\rangle|^2=|\psi_1(Q_1)\psi_3(Q_2)|^2 $. Where $\psi_i(Q)$ is the position wave function of a single harmonic oscillator with occupation number i. But I would expect the answer to be $|\psi_1(Q_1)\psi_3(Q_2)+\psi_3(Q_1)\psi_1(Q_2)|^2$ becaue of the exchange symmetry. What is my wrongthinking here? $\endgroup$ – curio Aug 27 '18 at 18:29
  • $\begingroup$ I don't see why we would particularly care about the exchange of two phonons between different modes since a priori the phonon is just a concept that we "made up" to make sense of the form of the hamiltonian isn't it? Before we diagonalized to hamiltonian, the state of the system was discribed by a sum of tensor products of single particle states which has to obey the exchange symmetry depending on whether the particles in question are bosons or fermions. Why don't we have to care about that anymore ? $\endgroup$ – curio Aug 27 '18 at 19:11

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