Many-body Quantum Mechanics¶

In this notebook, we briefly discuss the formalism of many-body theory from the point of view of quantum mechanics.

Table of Contents¶

A system of $N$ particles is described a wavefunction of $N$ coordinates – one for each particle. We also introduce the notion of the local $N$-body density:

$$ \Psi(r_1, r_2, \cdots, r_N), \qquad n_{(N)}(r_1, r_2, \cdots, r_N) = \Psi^*(r_1, r_2, \cdots, r_N)\Psi(r_1, r_2, \cdots, r_N). $$

To simplify notations, we shall often write $n_{(N)}(\{r_i\})$ where $\{r_i\}\equiv \{r_i\}_{i=1}^{N}$ represents the set of arguments. The normalization convention is

$$ \int \prod_{i}\d{r_i}\; n_{(N)}(\{r_i\}) = 1. $$

In operator notation, we represent the state $\ket{\Psi}$ as living in a product space. Individual operators act on their appropriate part of the space:

$$ \op{r}_1 = \op{r}\otimes\op{1}\otimes\cdots\otimes\op{1}\\ \op{r}_2 = \op{1}\otimes\op{r}\otimes\cdots\otimes\op{1}\\ \vdots\\ \op{r}_N = \op{1}\otimes\op{1}\otimes\cdots\otimes\op{r}. $$

As these operators all commute, they are simultaneously diagonalized by the eigenstates in the position basis $\ket{r_1, r_2, \dots, r_N}$ and we have

$$ \Psi(r_1, r_2, \cdots, r_N) = \braket{r_1, r_2, \dots, r_N|\Psi}. $$

The interpretation of this is that the probability distribution for the position of particle 1 (irrespective of where the other particles are) is:

$$ n_{(1)}(r) = \int \prod_{i=2}^{N}\d{r_i}\; n_{(N)}(r, \{r_i\}_{i=2}^{N}) = \int \prod_{i=1}^{N}\d{r_i}\; n_{(N)}(\{r_i\})\delta(r_1-r) = \braket{\delta(\op{r}_1-r)}. $$

Bosons and Fermions¶

The previous discussion has been for wavefunctions with distinguishable particles – for example, particles that have different charges. If one wants to discuss identical particles, then the wavefunction must either be even or odd under exchange, corresponding to bosons and fermions respectively. For bosons we have

$$ \Psi_B(r_1, r_2, \cdots, r_N) = \Psi_B(r_2, r_1, \cdots, r_N) $$

etc. for all permutations, while for fermions we have

$$ \Psi_F(r_1, r_2, \cdots, r_N) = -\Psi_F(r_2, r_1, \cdots, r_N) = \frac{1}{N!}\sum_{ab\cdots}\epsilon_{ab\cdots }\Psi_F(r_a, r_b, \cdots) $$

where $\epsilon_{abc\cdots}$ is the $N$-dimensional Levi-Civita symbol. Note that in either case, the density matrix is symmetric:

$$ \rho_{(N)}(r_1, r_2, \cdots, r_N) = \rho_{(N)}(r_2, r_1, \cdots, r_N). $$

This allows us to uniquely define a tower of "density matrices" recursively:

$$ n_{(N-1)}\Bigl(\{r_i\}_{i=1}^{N-1}\Bigr) = \int \d{r_{N}}\; n_{(N)}\Bigl(\{r_i\}_{i=1}^{N}\Bigr). $$

For identical particles we have a new notion of the "total density" normalized to the total number of particles $N$ in the system:

$$ n(r) = \sum_{i=1}^{N} \braket{\delta(\op{r}_i - r)} = N n_{(1)}(r). $$

Similarly, we can define the two-body density:

$$ n(r_1, r_2) = \sum_{i\neq j} \braket{\delta(\op{r}_i - r_1)\delta(\op{r}_j - r_2)} = N(N-1) n_{(2)}(r_1, r_2). $$

The terms corresponding to $i=j$ do not contribute for fermions, but must be excluded for bosons as we shall see below.

Non-interacting Particles¶

As a simple example, and to be concrete, we consider two non-interacting particles with each particle in a different single-particle state $\phi_{i}(r)$:

$$ \Psi^{B/F}(r_1,r_2) = \frac{1}{\sqrt{2}}\Bigl( \phi_1(r_1)\phi_2(r_2) \pm \phi_2(r_1)\phi_1(r_2) \Bigr),\\ n^{B/F}_{(2)}(r_1,r_2) = \frac{1}{2}\Bigl( \rho_1(r_1)\rho_2(r_2) + \rho_1(r_2)\rho_2(r_1) \pm \bigl( \rho_1(r_1,r_2)\rho_2(r_2,r_1) + \text{h.c.}\bigr) \Bigr),\\ n^{B/F}_{(1)}(r) = \int\d{r'}n^{B/F}_{(2)}(r,r') = \frac{\rho_1(r) + \rho_2(r)}{2}, $$

where $\rho_i(r_1,r_2) = \phi_i^*(r_1)\phi_i(r_2)$ is the non-local one-body density and $\rho_i(r) = \rho_i(r,r)$ is the local density. The orthonormality of the states $\int \d{r} \phi_i^*(r)\phi_j(r) = \delta_{ij}$ ensures the second relationship (the second $\pm$ term vanishes). For the final relationship to hold, we see why for bosons we must exclude the diagonal terms from the sum. (For fermions, they vanish automatically):

$$ n^{B/F}(r_1, r_2) = n^{B/F}_{(2)}(r_1,r_2) + n^{B/F}_{(2)}(r_2,r_1) = 2n^{B/F}_{(2)}(r_1,r_2). $$

For two bosons in the same state:

$$ \Psi^{B}(r_1,r_2) = \phi(r_1)\phi(r_2),\\ n^{B}_{(2)}(r_1,r_2) = \rho(r_1)\rho(r_2),\\ n^{B}_{(1)}(r) = \int\d{r'}n^{B}_{(2)}(r,r') = \rho(r),\\ n^{B}(r_1, r_2) = n^{B}_{(2)}(r_1,r_2) + n^{B}_{(2)}(r_2,r_1) = 2\rho(r_1)\rho(r_2). $$

The generalization for Fermions is best presented in terms of the generalized Kronecker delta:

$$ \epsilon_{a_1\cdots a_N}\epsilon_{b_1\cdots b_N} = N!\delta_{[a_1}^{b_1}\ldots\delta_{a_N]}^{b_N} = \delta_{a_1\cdots a_N}^{b_1\cdots b_N} $$

where $[a_1\cdots a_N]$ means the anti-symmetrized average over the indices, i.e.

$$ \delta_{[a_1}^{b_1}\delta_{a_2]}^{b_2} = \frac{1}{2}\left(\delta_{a_1}^{b_1}\delta_{a_2}^{b_2} - \delta_{a_2}^{b_1}\delta_{a_1}^{b_2}\right). $$\begin{align} \Psi(\{r_i\}) &= \frac{1}{\sqrt{N!}}\sum_{a_1a_2\cdots a_N}\epsilon_{a_1a_2\cdots a_N} \phi_{1}(r_{a_1})\phi_{2}(r_{a_2})\cdots\phi_{N}k(r_{a_N})\\ &= \frac{1}{\sqrt{N!}}\sum_{a_1a_2\cdots a_N}\epsilon_{a_1a_2\cdots a_N} \phi_{a_1}(r_1)\phi_{a_2}(r_2)\cdots\phi_{a_N}k(r_N)\\ n_{(N)}(\{r_i\}) &= \frac{1}{N!}\sum_{a_1a_2\cdots a_N;b_1b_2\cdots b_N} \delta_{a_1a_2\cdots a_N}^{b_1b_2\cdots b_N} \rho_{a_1b_1}(r_1)\rho_{a_2b_2}(r_2)\cdots\rho_{a_Nb_N}(r_N)\\ n_{(N-1)}(\{r_i\}^{N-1}) &= \frac{1}{N!}\sum_{a_1a_2\cdots a_{N-1};b_1b_2\cdots b_{N-1}}\sum_{a_N} \delta_{a_1a_2\cdots a_N}^{b_1b_2\cdots a_N} \rho_{a_1b_1}(r_1)\rho_{a_2b_2}(r_2)\cdots\rho_{a_{N-1}b_{N-1}}(r_{N-1})\\ n_{(2)}(r_1, r_2) &= \frac{1}{N!}\sum_{a_1a_2;b_1b_2}\sum_{a_3\cdots a_N} \delta_{a_1a_2a_3\cdots a_N}^{b_1b_2a_3\cdots a_N} \rho_{a_1b_1}(r_1)\rho_{a_2b_2}(r_2)\\ &= \frac{1}{N(N-1)}\sum_{a_1a_2;b_1b_2}\delta_{a_1a_2}^{b_1b_2} \rho_{a_1b_1}(r_1)\rho_{a_2b_2}(r_2)\\ &= \frac{1}{N(N-1)}\sum_{a_1a_2}\Bigl( \rho_{a_1}(r_1)\rho_{a_2}(r_2) - \rho_{a_1a_2}(r_1)\rho_{a_2a_1}(r_2)\Bigr)\\ &= \frac{1}{N(N-1)}\sum_{a_1a_2}\Bigl( \rho_{a_1}(r_1)\rho_{a_2}(r_2) - \rho_{a_1}(r_1,r_2)\rho_{a_2}(r_2,r_1)\Bigr) \tag{direct and exchange}\\ n_{(1)}(r) &= \frac{1}{N!}\sum_{a_1;b_1}\sum_{a_2\cdots a_N} \delta_{a_1a_2\cdots a_N}^{b_1a_2\cdots a_N} \rho_{a_1b_1}(r) \\ &= \frac{1}{N}\sum_{a}\rho_{a}(r). \end{align}

where $\rho_{ii'}(r) = \phi_i^*(r)\phi_i'(r)$ and we have used the relations $\int\d{r_N}\rho_{kk'}(r_N) = \delta_{kk'}$ and $\sum_{a_1\cdots a_k}\delta_{a_1\cdots a_k a_{k+1}\cdots a_N}^{a_1\cdots a_k b_{k+1}\cdots b_N} = k!\delta_{a_{k+1}\cdots a_N}^{b_{k+1}\cdots b_N}$.

Note: The most common normalization convention for the density matrices are $N n_{(1)}$ and $N(N-1)n_{(2)}$. We have not yet adopted this here.