Abstract Generated abstract
This note applies Zubarev’s method of additional variables to the statistical sum of a system of interacting particles, formulated explicitly for fermions with a two-particle central potential. After a unitary transformation, the Hamiltonian is separated into free-particle, oscillator, and interaction parts, and the calculation is carried out in the approximation that neglects the residual interaction term. The authors derive an expression for the approximate statistical sum in terms of particle coordinates and oscillator eigenfunctions, perform the oscillator summations using a Hermite-function identity, and obtain a simplified upper estimate by neglecting antisymmetrization. For large particle number they further evaluate the resulting configurational integral, yielding a product formula involving the oscillator frequencies and coefficients determined by the Fourier transform of the interaction potential.
Full Text
Physics
A. Pavlikovskii and V. Shurubna
On the Application of Zubarev’s Method of Additional Variables to Statistical Physics
(Presented by Academician N. N. Bogolyubov, 16 VIII 1957)
As is well known, the problem of investigating the physical properties of dynamical systems consisting of a large number of interacting particles is very complicated. In 1953 D. N. Zubarev, in connection with the problem of elementary excitations in a real Fermi gas, developed the method of additional variables and applied it to the calculation of the energy spectrum of these excitations.
In the present note the method of additional variables is applied to the calculation of the statistical sum of a system of \(N\) interacting particles. The statistical sum \(Z_N\) is the trace of the operator \(e^{-\beta \hat H}\), where \(\beta = \frac{1}{kT}\), and \(\hat H\) is the Hamiltonian of the system.
For definiteness, let us consider Fermi particles (the consideration of Bose particles is entirely analogous) in a volume \(V\), interacting by means of a two-particle central potential \(W_{ij}=W(|\mathbf{x}_i-\mathbf{x}_j|)\). The Hamiltonian of the system is represented in the form
\[ \hat H=\hat H_0+\hat H_1, \tag{1} \]
where
\[ \hat H_0=\sum_{i=1}^{N}\frac{\hat p_i^{\,2}}{2m},\qquad \hat H_1=\frac{1}{2}\sum_{\substack{i,j\\ i\ne j}}\hat W_{ij}. \]
We now introduce additional operators* \(\hat Q_{\mathbf{k}}\) \((|\mathbf{k}|<k_0,\ \mathbf{k}\ne 0,\ \hat Q_{\mathbf{k}}^{+}=\hat Q_{-\mathbf{k}},\ +\) denotes Hermitian conjugation). The state vector of the system satisfies the additional conditions
\[ \hat Q_{\mathbf{k}}|\Psi\rangle=0\qquad (|\mathbf{k}|<k_0). \tag{2} \]
We have
\[ Z_N=S_{\Psi}\langle\Psi|e^{-\beta\hat H}|\Psi\rangle =S_{\Phi}\langle\Phi|e^{-\beta\hat H'}|\Phi\rangle, \tag{3} \]
where
\[ |\Phi\rangle=\hat U|\Psi\rangle,\qquad \hat H'=\hat U\hat H\hat U^{-1},\qquad \hat U^{+}=\hat U^{-1}. \]
We take
\[ \hat U=\exp\left\{-\frac{i}{\hbar}\sum_{|\mathbf{k}|<k_0}^{\prime}\hat P_{\mathbf{k}}\hat\rho_{\mathbf{k}}\right\}. \tag{4} \]
* We use the Hermitian variant of Zubarev’s method \((^{1})\).
\(\sum'\) denotes summation over \(\mathbf{k}\ne 0\); \(\hat P_{\mathbf{k}}\) is the operator canonically conjugate to \(\hat Q_{\mathbf{k}}\);
\[ \hat\rho_{\mathbf{k}}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N} e^{-i\mathbf{k}\hat{\mathbf{x}}_j}, \qquad \hat\rho_{\mathbf{k}}=\hat\rho_{\mathbf{k}}^{R}+i\hat\rho_{\mathbf{k}}^{I}. \]
The vectors \(|\Phi\rangle\) satisfy the conditions
\[ (\hat Q_{\mathbf{k}}-\hat\rho_{\mathbf{k}})|\Phi\rangle=0 \qquad (|\mathbf{k}|<k_0). \tag{5} \]
The Hamiltonian is
\[ \hat H'=\hat H'_0+\hat H'_{osc}+\hat H'_I, \tag{6} \]
where
\[ \hat H'_0=\sum_{i=1}^{N}\frac{\hat{\mathbf{p}}_i^{\,2}}{2m} \]
is the Hamiltonian of “free” fermions;
\[ \hat H'_{osc}=\frac{1}{2m}\sum_{|\mathbf{k}|<k_0}' k^2\hat P_{\mathbf{k}}\hat P_{-\mathbf{k}} +\frac{N}{2V}\sum_{|\mathbf{k}|<k_0}' W_k\hat Q_{\mathbf{k}}\hat Q_{-\mathbf{k}} . \]
is the Hamiltonian of “free” oscillators, whose vibration frequencies are
\[ \omega_k= \sqrt{\frac{1}{m}\frac{N}{V}\,k^2 W_k}, \qquad W_k=\int W(|\mathbf{x}|)e^{-i\mathbf{k}\mathbf{x}}\,d^3x; \]
\(\hat H'_I\) contains the interaction terms between fermions, between oscillators, and between fermions and oscillators \((^1)\).
We shall be interested in the approximation in which the term \(\hat H'_I\) is neglected.*
To compute the trace in formula (3), we put
\[ |\Phi\rangle=\sum_{x,Q}\langle x,Q|\Phi\rangle |x,Q\rangle, \tag{7} \]
where
\[ |x,Q\rangle\equiv \prod_{m=1}^{N}|x_m\rangle \prod_{|\mathbf{k}|<k_0}'' |Q_{\mathbf{k}}\rangle^{**} \]
is a common eigenvector of the operators \(\hat{\mathbf{x}}_1,\ldots,\hat{\mathbf{x}}_N\) and \(\hat Q_{\mathbf{k}}\) \((|\mathbf{k}|<k_0)\).
On the basis of (5),
\[ (\hat Q_{\mathbf{k}}-\hat\rho_{\mathbf{k}})|\Phi\rangle \equiv \sum_{x,Q}\langle x,Q|\Phi\rangle \bigl(Q_{\mathbf{k}}-\rho_{\mathbf{k}}(\mathbf{x}_1,\ldots,\mathbf{x}_N)\bigr)|x,Q\rangle=0, \tag{8} \]
whence
\[ \langle x,Q|\Phi\rangle \bigl(Q_{\mathbf{k}}-\rho_{\mathbf{k}}(\mathbf{x}_1,\ldots,\mathbf{x}_N)\bigr)=0. \tag{9} \]
* Usually, in the method of elementary excitations, it is assumed that this approximation is good at low temperatures. But at low temperatures an approximation to the Hamiltonian is good in which the lowest energy levels are well approximated. It is accepted without proof that the approximation under consideration has this property. This observation is due to Prof. R. S. Ingarden, to whom we express our gratitude.
** \(|Q_{\mathbf{k}}\rangle=|Q_{\mathbf{k}}^{R}\rangle |Q_{\mathbf{k}}^{I}\rangle\), \(\hat Q_{\mathbf{k}}=\hat Q_{\mathbf{k}}^{R}+i\hat Q_{\mathbf{k}}^{I}\); \(\prod''\) means that the product is taken over \(\mathbf{k}\) \((\mathbf{k}\ne0)\) lying in a hemisphere, since \(Q_{-\mathbf{k}}^{R}=Q_{\mathbf{k}}^{R}\), \(Q_{-\mathbf{k}}^{I}=-Q_{\mathbf{k}}^{I}\).
Therefore an arbitrary vector \(|\Phi\rangle\), satisfying the supplementary conditions (5), can be expanded in the complete system of vectors of the type \(|x, Q=\rho(x)\rangle\), and
\[ \operatorname*{S}_{\Phi}\langle \Phi|\ldots|\Phi\rangle = \operatorname*{S}_{x}\langle x, Q=\rho(x)|\ldots|x, Q=\rho(x)\rangle . \tag{10} \]
For the statistical sum in the approximation under consideration we have
\[ Z_N^0=\int d^3x_1\ldots d^3x_N \langle x|e^{-\beta \hat H'_0}|x\rangle \langle Q=\rho(x)|e^{-\beta \hat H'_{osc}}|Q=\rho(x)\rangle \tag{11} \]
and after carrying out the calculations we obtain
\[ Z_N^0= \left(\frac{1}{N!}\right)^2 V^{-N} \sum_{\mathbf f_1}\ldots\sum_{\mathbf f_N} \exp\left\{-\beta\sum_{j=1}^{N}\varepsilon_{\mathbf f_j}\right\} \int d^3x_1\ldots d^3x_N \left\|e^{i\mathbf f_a\mathbf x_b}\right\|^2 \times \]
\[ \times \prod_{|\mathbf k|<k_0}^{\prime\prime} \sum_{m_{\mathbf k}=0}^{\infty} \sum_{l_{\mathbf k}=0}^{\infty} e^{-2\beta\hbar\omega_k(m_{\mathbf k}+l_{\mathbf k}+1)} \left|\chi_{m_{\mathbf k}}\!\left(\rho_{\mathbf k}^R(x)\right)\right|^2 \left|\chi_{l_{\mathbf k}}\!\left(\rho_{\mathbf k}^I(x)\right)\right|^2 ; \tag{12} \]
here \(\varepsilon_{\mathbf f}=\mathbf f^2/2m\); \(\left\|e^{i\mathbf f_a\mathbf x_b}\right\|^2\) is the square of the modulus of the determinant \((a,b=1,\ldots,N)\);
\[ \chi_m(y)=\frac{1}{\sqrt c}\, e^{-\frac12\left(\frac{y}{c}\right)^2}\, \mathcal H_m\!\left(\frac{y}{c}\right) \]
\[ \left( c=\sqrt{\frac{1}{\hbar}}\, \sqrt[4]{\frac{1}{m}\,\frac{V}{N}\,\frac{k^2}{W_k}}, \quad \mathcal H_m(y)= \frac{(-1)^m}{\sqrt{2^m m!\sqrt\pi}}\, e^{y^2}\frac{d^n}{dy^n}e^{-y^2} \text{ — normalized Hermite polynomial} \right). \]
Formula (²)
\[ \sum_{m=0}^{\infty}[\chi_m(y)]^2 e^{-\alpha m} = \frac{1}{\sqrt\pi\,c\sqrt{1-e^{-2\alpha}}}\, e^{-\frac{1}{c^2}\operatorname{tgh}\frac{\alpha}{2}\,y^2} \tag{13} \]
allows the summations over \(m_{\mathbf k}\) and \(l_{\mathbf k}\) in (12) to be performed. Then
\[ Z_N^0= \exp\left\{-\beta\sum_{|\mathbf k|<k_0}^{\prime}\hbar\omega_k\right\} \left(\frac{1}{N!}\right)^2 V^{-N} \times \]
\[ \times \sum_{\mathbf f_1}\ldots\sum_{\mathbf f_N} \exp\left\{-\beta\sum_{j=1}^{N}\varepsilon_{\mathbf f_j}\right\} \int d^3x_1\ldots d^3x_N \left\|e^{i\mathbf f_a\mathbf x_b}\right\|^2 \prod_{|\mathbf k|<k_0}^{\prime\prime} B_k e^{-A_k\rho_{\mathbf k}(x)\rho_{-\mathbf k}(x)}, \tag{14} \]
where
\[ A_k=\frac{1}{\hbar} \sqrt{m\frac{N}{V}\frac{W_k}{k^2}}\, \operatorname{tgh}\beta\hbar\omega_k, \qquad B_k=\frac{1}{\hbar\Pi} \sqrt{m\frac{N}{V}\frac{W_k}{k^2}\, \frac{1}{1-e^{-4\beta\hbar\omega_k}}}. \tag{15} \]
The calculation of the integral over \(\mathbf x_1,\ldots,\mathbf x_N\) is difficult. We shall calculate this integral by replacing
\(\left(\frac{1}{N!}\right)^2\left\|e^{i\mathbf f_a\mathbf x_b}\right\|^2\) by unity, which corresponds to neglecting the antisymmetrization of the states. This gives an estimate of the value of \(Z_N^0\) from above*:
\[ Z_N^{0\max} = \sum_{\mathbf f_1}\ldots\sum_{\mathbf f_N} \exp\left\{-\beta\sum_{j=1}^{N}\varepsilon_{\mathbf f_j}\right\} R_N, \tag{16} \]
* Z. Galiasevich drew our attention to this.
where
\[ R_N=\exp\left\{-\beta\sum_{|\mathbf{k}|<k_0}^{\prime}\hbar\omega_k\right\} V^{-N}\prod_{|\mathbf{k}|<k_0}^{\prime\prime} B_k \int d^3x_1\ldots d^3x_N \prod_{|\mathbf{k}|<k_0}^{\prime\prime} e^{-A_k\rho_k\rho_{-\mathbf{k}}}; \tag{17} \]
this is the configurational integral.
Let us compute \(R_N\) approximately for large \(N\), following exactly Zubarev’s method from paper \((^3)\). It is not difficult to verify that in this approximation
\[ R_N=\exp\left\{-\beta\sum_{|\mathbf{k}|<k_0}\hbar\omega_k\right\} \prod_{|\mathbf{k}|<k_0}^{\prime\prime} B_k\,(A_k+1)^{-1}. \tag{18} \]
The connection of the present result with the classical case of this problem will be considered in the following paper.
Institute of Physics
Polish Academy of Sciences,
Wrocław
Received
15 VIII 1957
CITED LITERATURE
1 D. N. Zubarev, Candidate dissertation, Moscow, 1953; ZhETF, 25, 548 (1953); Z. Galasiewicz, Postępy fizyki, 7, 317 (1956). 2 E. C. Titchmarsh, Introduction to the Theory of the Fourier Integrals, Oxford, 1948. 3 D. N. Zubarev, DAN, 95, 757 (1954).