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Kernel: Python [default]

Many-body Quantum Mechanics

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

import mmf_setup;mmf_setup.nbinit()
<IPython.core.display.Javascript object>

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

Ψ(r1,r2,⋯ ,rN),n(N)(r1,r2,⋯ ,rN)=Ψ∗(r1,r2,⋯ ,rN)Ψ(r1,r2,⋯ ,rN).\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)({ri})n_{(N)}(\{r_i\}) where {ri}≡{ri}i=1N\{r_i\}\equiv \{r_i\}_{i=1}^{N} represents the set of arguments. The normalization convention is

∫∏idri  n(N)({ri})=1.\int \prod_{i}\d{r_i}\; n_{(N)}(\{r_i\}) = 1.

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

ParseError: KaTeX parse error: Undefined control sequence: \op at position 1: \̲o̲p̲{r}_1 = \op{r}\…

As these operators all commute, they are simultaneously diagonalized by the eigenstates in the position basis ∣r1,r2,…,rN⟩\ket{r_1, r_2, \dots, r_N} and we have

Ψ(r1,r2,⋯ ,rN)=⟨r1,r2,…,rN∣Ψ⟩.\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:

ParseError: KaTeX parse error: Undefined control sequence: \op at position 166: …\braket{\delta(\̲o̲p̲{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

ΨB(r1,r2,⋯ ,rN)=ΨB(r2,r1,⋯ ,rN)\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

ΨF(r1,r2,⋯ ,rN)=−ΨF(r2,r1,⋯ ,rN)=1N!∑ab⋯ϵab⋯ΨF(ra,rb,⋯ )\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 ϵabc⋯\epsilon_{abc\cdots} is the NN-dimensional Levi-Civita symbol. Note that in either case, the density matrix is symmetric:

ρ(N)(r1,r2,⋯ ,rN)=ρ(N)(r2,r1,⋯ ,rN).\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)({ri}i=1N−1)=∫drN  n(N)({ri}i=1N).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 NN in the system:

ParseError: KaTeX parse error: Undefined control sequence: \op at position 38: …\braket{\delta(\̲o̲p̲{r}_i - r)} = N…

Similarly, we can define the two-body density:

ParseError: KaTeX parse error: Undefined control sequence: \op at position 45: …\braket{\delta(\̲o̲p̲{r}_i - r_1)\de…

The terms corresponding to i=ji=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 ϕi(r)\phi_{i}(r):

ΨB/F(r1,r2)=12(ϕ1(r1)ϕ2(r2)±ϕ2(r1)ϕ1(r2)),n(2)B/F(r1,r2)=12(ρ1(r1)ρ2(r2)+ρ1(r2)ρ2(r1)±(ρ1(r1,r2)ρ2(r2,r1)+h.c.)),n(1)B/F(r)=∫dr′n(2)B/F(r,r′)=ρ1(r)+ρ2(r)2,\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 ρi(r1,r2)=ϕi∗(r1)ϕi(r2)\rho_i(r_1,r_2) = \phi_i^*(r_1)\phi_i(r_2) is the non-local one-body density and ρi(r)=ρi(r,r)\rho_i(r) = \rho_i(r,r) is the local density. The orthonormality of the states ∫drϕi∗(r)ϕj(r)=δij\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):

nB/F(r1,r2)=n(2)B/F(r1,r2)+n(2)B/F(r2,r1)=2n(2)B/F(r1,r2).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:

ΨB(r1,r2)=ϕ(r1)ϕ(r2),n(2)B(r1,r2)=ρ(r1)ρ(r2),n(1)B(r)=∫dr′n(2)B(r,r′)=ρ(r),nB(r1,r2)=n(2)B(r1,r2)+n(2)B(r2,r1)=2ρ(r1)ρ(r2).\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:

ϵa1⋯aNϵb1⋯bN=N!δ[a1b1…δaN]bN=δa1⋯aNb1⋯bN\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 [a1⋯aN][a_1\cdots a_N] means the anti-symmetrized average over the indices, i.e.

δ[a1b1δa2]b2=12(δa1b1δa2b2−δa2b1δa1b2).\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).Ψ({ri})=1N!∑a1a2⋯aNϵa1a2⋯aNϕ1(ra1)ϕ2(ra2)⋯ϕNk(raN)=1N!∑a1a2⋯aNϵa1a2⋯aNϕa1(r1)ϕa2(r2)⋯ϕaNk(rN)n(N)({ri})=1N!∑a1a2⋯aN;b1b2⋯bNδa1a2⋯aNb1b2⋯bNρa1b1(r1)ρa2b2(r2)⋯ρaNbN(rN)n(N−1)({ri}N−1)=1N!∑a1a2⋯aN−1;b1b2⋯bN−1∑aNδa1a2⋯aNb1b2⋯aNρa1b1(r1)ρa2b2(r2)⋯ρaN−1bN−1(rN−1)n(2)(r1,r2)=1N!∑a1a2;b1b2∑a3⋯aNδa1a2a3⋯aNb1b2a3⋯aNρa1b1(r1)ρa2b2(r2)=1N(N−1)∑a1a2;b1b2δa1a2b1b2ρa1b1(r1)ρa2b2(r2)=1N(N−1)∑a1a2(ρa1(r1)ρa2(r2)−ρa1a2(r1)ρa2a1(r2))=1N(N−1)∑a1a2(ρa1(r1)ρa2(r2)−ρa1(r1,r2)ρa2(r2,r1))n(1)(r)=1N!∑a1;b1∑a2⋯aNδa1a2⋯aNb1a2⋯aNρa1b1(r)=1N∑aρa(r).\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 ρii′(r)=ϕi∗(r)ϕi′(r)\rho_{ii'}(r) = \phi_i^*(r)\phi_i'(r) and we have used the relations ∫drNρkk′(rN)=δkk′\int\d{r_N}\rho_{kk'}(r_N) = \delta_{kk'} and ∑a1⋯akδa1⋯akak+1⋯aNa1⋯akbk+1⋯bN=k!δak+1⋯aNbk+1⋯bN\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 Nn(1)N n_{(1)} and N(N−1)n(2)N(N-1)n_{(2)}. We have not yet adopted this here.