shen's notes on advanced quantum mechanics
2895044375@qq.com
reference books :
高等量子力学，第二版，喀兴林，高等教育出版社，ISBN: 9787040099256
量子力学，第四版，卷II，曾谨言，科学出版社，ISBN: 9787030190215
C1 -- Mathematical Fundamentals and Physical Principles
1.1 Basic principles of quantum mechanics
(1) state of a microsystem <--> vector in a Hilbert space
\[|\Psi\rangle = \sum_{i=1}^N c_i|\psi_i\rangle
\]
\[|\Psi\rangle := [c_1, c_2, ..., c_N]^\top, \langle\Psi| := [c_1, c_2, ..., c_N]
\]
(2) physical quantity of a microsystem <--> Hermite operatror in Hilbert space
\[\hat{O}|\psi_i\rangle = \lambda_i|\psi_i\rangle
\]
\[\langle\hat{O}\rangle = \langle\Psi|\hat{O}|\Psi\rangle
= \langle\sum_{i=1}^Nc_i\psi_i|\hat{O}|\sum_{j=1}^Nc_j\psi_j\rangle
= \sum_{ij}c_ic_j\langle\psi_i|\hat{O}|\psi_j\rangle
= \sum_{ij}c_ic_j\lambda_i\delta_{ij} = \sum_i|c_i|^2\lambda_i\]
(3) commutation between location operator and momentum operator
\[[\mathbf{x_i}, \mathbf{x_j}] = [\mathbf{p_i}, \mathbf{p_j}] = 0
\]
\[[\mathbf{x_i}, \mathbf{p_j}] = i\hbar\delta_{ij}
\]
(4) evolution of the state of a microsystem --> Schr\(\ddot{\text{o}}\)dinger equation
\[i\hbar\frac{\partial|\Psi(t)\rangle}{\partial t} = \hat{H}|\Psi(t)\rangle
\]
(5) symmetry of identical particles --> symmetrical (Boson), antisymmetrical (Fermi)
\[|\Psi(..., \mathbf{q_i}, ..., \mathbf{q_j}, ...)\rangle = \pm|\Psi(..., \mathbf{q_j}, ..., \mathbf{q_i}, ...)\rangle
\]
1.2 Hilbert space
vector space : a set \(\mathcal{H} = \{\psi. \phi, ...\}\) where all elements satisfy the rules
(1) additon --> \(\forall \psi, \phi \in \mathcal{H}\), \(\chi = \psi + \phi \in \mathcal{H}\)
(2) multiplication --> \(\forall a\in\mathbb{C}\) and \(\psi\in\mathcal{H}\), \(\chi = a\psi\in\mathcal{H}\)
(3) inner product --> \(\forall \psi,\phi\in\mathcal{H}\), \((\psi, \phi) = C \in \mathbb{C}\)
basis vector : an orthogonal complete set of the vector space
projection onto basis vector \(|\psi_i\rangle\) --> \(|\psi_i\rangle\langle\psi_i|\)
--> \(\sum_i|\psi_i\rangle\langle\psi_i| = 1\)
\[\langle\Phi|\Psi\rangle = \left(\sum_i\langle\Phi|\psi_i\rangle\langle\psi_i|\right)|\Psi\rangle
= \sum_i\langle\Phi|\psi_i\rangle\langle\psi_i|\Psi\rangle\]
complete space : for all Cauchy sequence \(\{S_n\}\) in the vector space, \(\{S_n\}\) converges to a vector in the space
Cauchy sequence \(\{S_n\}\) --> \(\forall\varepsilon>0, \exists N, s.t. \forall m,n>N\Rightarrow|S_n-S_m|<\varepsilon\)
Hilbert space : infinite-dimensional complete vector space
conjugate space : ket --> \(|\psi\rangle \in \mathcal{H}\), bra --> \(\langle\psi| \in \mathcal{H}^*\) --> \(\langle\psi| = |\psi\rangle^*\)
1.3 Operator
operator : transformation between vectors in Hilbert space
\[\forall |\psi\rangle \in \mathcal{H}, \hat{O}|\psi\rangle = |\phi\rangle \in \mathcal{H}
\]
\[\hat{O} \{|\psi\rangle+|\phi\rangle\} = \hat{O}|\psi\rangle + \hat{O}|\phi\rangle
\]
conjugate of an operator -->
\(\langle\psi|\hat{A}^\dagger|\phi\rangle = (\langle\psi|\hat{A}^\dagger)|\phi\rangle
= (\hat{A}|\psi\rangle)^*(\langle\phi|)^* = \langle\phi|\hat{A}|\psi\rangle^*\)
Harmite operator --> \(\hat{A}^\dagger = \hat{A}\)
unitrary operator --> \(\hat{U}^{-1} = \hat{U}^\dagger\)
eigenvalue
\[\hat{A}|\psi_i\rangle = a_i|\psi_i\rangle, \hat{A}\langle\psi_i| = a_i'\langle\psi_i|
\]
linear operator --> left and right eigenvalue spectrums are identical
Hermite operator --> eigenvalues are real, eigenvectors of different eigenvalues are orthogonal
projective operator --> \(\hat{P_S}|\psi\rangle = |\psi_S\rangle \in \mathcal{S} \subset \mathcal{H}, \forall |\psi\rangle \in\mathcal{H}\)
(1) idempotent --> \(\hat{P_S}^2 = \hat{P_S}\)
Proof :
\[\hat{P_S}^2|\psi\rangle = \hat{P_S}|\psi_S\rangle = |\psi_S\rangle
\]
(2) eigenvalues of \(\hat{P_S}\) are 0 and 1, eigenvectors span the whole Hilbert space \(\mathcal{H} = \mathcal{S} + \mathcal{S'}\)
Proof :
\[\hat{P_S}|\psi_S\rangle = |\psi_S\rangle \Rightarrow (\hat{P_S}-1)|\psi_S\rangle = 0,
\forall|\psi_S\rangle\in\mathcal{S}\]
\[\hat{P_S}|\psi_{S'}\rangle = 0 \Rightarrow (\hat{P_S}-0)|\psi_{S'}\rangle = 0,
\forall|\psi_{S'}\rangle\in\mathcal{S'}\]
\[|\psi\rangle = |\psi_S\rangle + |\psi_{S'}\rangle, \forall|\psi\rangle\in\mathcal{H}
\]
(3) \(\hat{P} = |\psi_i\rangle\langle\psi_i|\) is a projective operator
Proof :
\[\forall |\Psi\rangle \in\mathcal{H}, \hat{P}|\Psi\rangle = |\psi_i\rangle\langle\psi_i|\Psi\rangle = c_i|\psi_i\rangle
\]
1.4 Representation theory
matrix representation --> in the basis \(\{|\psi_i\rangle\}\), we have
\(\diamondsuit\) ket
\[|\Psi\rangle = \sum_i|\psi_i\rangle\langle\psi_i|\Psi\rangle
= \begin{bmatrix}|\psi_1\rangle & |\psi_2\rangle & \dots & |\psi_N\rangle\end{bmatrix}
\begin{bmatrix}c_1 \\ c_2 \\ \vdots \\ c_N\end{bmatrix}
:= \begin{bmatrix}c_1 \\ c_2 \\ \vdots \\ c_N\end{bmatrix}\]
\(\heartsuit\) bra
\[\langle\Psi| = \sum_i\langle\Psi|\psi_i\rangle\langle\psi_i|
= \begin{bmatrix}c_1^* & c_2^* & \dots & c_N^*\end{bmatrix}
\begin{bmatrix}|\psi_1\rangle \\ |\psi_2\rangle \\ \vdots \\ |\psi_N\rangle\end{bmatrix}
:= \begin{bmatrix}c_1^* & c_2^* & \dots & c_N^*\end{bmatrix}\]
\(\clubsuit\) inner product
\[\begin{align*}
\langle\Psi|\Phi\rangle
&= \sum_i\langle\Psi|\psi_i\rangle\langle\psi_i| \sum_j|\psi_j\rangle\langle\psi_j|\Phi\rangle
= \sum_{ij} \langle\Psi|\psi_i\rangle\langle\psi_i|\psi_j\rangle\langle\psi_j|\Phi\rangle \\
&= \sum_i \langle\Psi|\psi_i\rangle\langle\psi_i|\Phi\rangle
= \begin{bmatrix}c_1^* & c_2^* & \dots & c_N^*\end{bmatrix}
\begin{bmatrix}c_1 \\ c_2 \\ \vdots \\ c_N\end{bmatrix}
\end{align*}\]
\(\spadesuit\) operator
\[\begin{align*}
\hat{A}|\Psi\rangle
&= \sum_{ij}|\psi_i\rangle\langle\psi_i| \left(\hat{A}|\psi_j\rangle\langle\psi_j|\Psi\rangle\right)
= \sum_{ij}|\psi_i\rangle \left(\langle\psi_i|\hat{A}|\psi_j\rangle\right)\langle\psi_j|\Psi\rangle \\
&= \begin{bmatrix}|\psi_1\rangle & |\psi_2\rangle & \dots & |\psi_N\rangle\end{bmatrix}
\begin{bmatrix}
\langle\psi_1|\hat{A}|\psi_1\rangle & \langle\psi_1|\hat{A}|\psi_2\rangle & \dots & \langle\psi_1|\hat{A}|\psi_N\rangle \\
\langle\psi_2|\hat{A}|\psi_1\rangle & \langle\psi_2|\hat{A}|\psi_2\rangle & \dots & \langle\psi_2|\hat{A}|\psi_N\rangle \\
\vdots & \vdots & \ddots & \vdots\\
\langle\psi_N|\hat{A}|\psi_1\rangle & \langle\psi_N|\hat{A}|\psi_2\rangle & \dots & \langle\psi_N|\hat{A}|\psi_N\rangle \\
\end{bmatrix}
\begin{bmatrix}c_1 \\ c_2 \\ \vdots \\ c_N\end{bmatrix}
\end{align*}\]
representation transformation
in the basis \(\{|\psi_i\rangle\}\), we have
\[|\Psi\rangle = \sum_i|\psi_i\rangle\langle\psi_i|\Psi\rangle
:= \begin{bmatrix}c_1 & c_2 & \dots & c_N\end{bmatrix}^\top\]
in the basis \(\{|\phi_i\rangle\}\), we have
\[|\Psi\rangle = \sum_i|\phi_i\rangle\langle\phi_i|\Psi\rangle
:= \begin{bmatrix}b_1 & b_2 & \dots & b_N\end{bmatrix}^\top\]
transformation from basis \(\{|\phi_i\rangle\}\) to basis \(\{|\psi_i\rangle\}\)
\[\langle\psi_i|\Psi\rangle = \sum_j\langle\psi_i|\phi_j\rangle\langle\phi_j|\Psi\rangle
\]
\[\begin{bmatrix}c_1 \\ c_2 \\ \vdots \\ c_N\end{bmatrix}
= \begin{bmatrix}
\langle\psi_1|\phi_1\rangle & \langle\psi_1|\phi_2\rangle & \dots & \langle\psi_1|\phi_N\rangle \\
\langle\psi_2|\phi_1\rangle & \langle\psi_2|\phi_2\rangle & \dots & \langle\psi_2|\phi_N\rangle \\
\vdots & \vdots & \ddots & \vdots\\
\langle\psi_N|\phi_1\rangle & \langle\psi_N|\phi_2\rangle & \dots & \langle\psi_N|\phi_N\rangle \\
\end{bmatrix}
\begin{bmatrix}b_1 \\ b_2 \\ \vdots \\ b_N\end{bmatrix}
:= \hat{U}_{\phi\to\psi}\begin{bmatrix}b_1 \\ b_2 \\ \vdots \\ b_N\end{bmatrix}\]
\[\langle\psi_i|\hat{A}|\psi_j\rangle
= \sum_{kl}\langle\psi_i|\phi_k\rangle\langle\phi_k|\hat{A}|\phi_l\rangle\langle\phi_l|\psi_j\rangle\]
\[\begin{bmatrix}
\langle\psi_1|\hat{A}|\psi_1\rangle & \dots & \langle\psi_1|\hat{A}|\psi_N\rangle \\
\vdots & \ddots & \vdots\\
\langle\psi_N|\hat{A}|\psi_1\rangle & \dots & \langle\psi_N|\hat{A}|\psi_N\rangle \\
\end{bmatrix} :=
\hat{U}_{\phi\to\psi}
\begin{bmatrix}
\langle\phi_1|\hat{A}|\phi_1\rangle & \dots & \langle\phi_1|\hat{A}|\phi_N\rangle \\
\vdots & \ddots & \vdots\\
\langle\phi_N|\hat{A}|\phi_1\rangle & \dots & \langle\phi_N|\hat{A}|\phi_N\rangle \\
\end{bmatrix}
\hat{U}_{\psi\to\phi}
\]
axiom : representation transformations are unitary transformation
Proof :
\[\begin{bmatrix}b_1 \\ b_2 \\ \vdots \\ b_N\end{bmatrix}
= \hat{U}_{\phi\to\psi}^{-1} \begin{bmatrix}c_1 \\ c_2 \\ \vdots \\ c_N\end{bmatrix}
= \hat{U}_{\psi\to\phi} \begin{bmatrix}c_1 \\ c_2 \\ \vdots \\ c_N\end{bmatrix}\]
\[\hat{U}_{\psi\to\phi}^\dagger = [(\langle\phi_i|\psi_j\rangle)^*]^\top
= [(\langle\phi_j|\psi_i\rangle)^*] = [\langle\psi_i|\phi_j\rangle] = \hat{U}_{\phi\to\psi}\]
\[\hat{U}_{\psi\to\phi}^\dagger = \hat{U}_{\psi\to\phi}^{-1}
\]
C2 - Density Matrix and Complex System
2.1 Density operator
considering a complete set of commuting conserved observables \(F\) (observables in the set are commuting with each other, and their simultaneous eigenstates can form a complete basis for the system, e.g., (\(\hat{x}, \hat{y}, \hat{z}\)), (\(\hat{p}_x, \hat{p}_y, \hat{p}_z\))), we have the representation based on \(\{|\psi_i\rangle\}\) called \(F\) representation
completeness :
\[\sum_i |\psi_i\rangle\langle\psi_i| = 1
\]
quantum state in \(F\) representation
\[|\Psi\rangle = \sum_i|\psi_i\rangle\langle\psi_i|\Psi\rangle = \sum_i c_i|\psi_i\rangle
\]
density operator \(\hat{\rho} = |\Psi\rangle\langle\Psi|\) in \(F\) representation
\[\hat{\rho}_{\{|\psi_i\rangle\}} = \langle\psi_i|\hat{\rho}|\psi_j\rangle
= \langle\psi_i|\Psi\rangle \langle\Psi|\psi_j\rangle
= \begin{bmatrix}c_1 \\ c_2 \\ \vdots \\ c_N\end{bmatrix}
\begin{bmatrix}c_1 & c_2 & \dots & c_N\end{bmatrix}
\]
2.2 Mixed state
\(\diamondsuit\) for a pure state, we have the density operator
\[\hat{\rho} = |\Psi\rangle\langle\Psi|
\]
average of observable \(B\)
\[\langle\hat{B}\rangle = \langle\Psi|\hat{B}|\Psi\rangle
= \sum_i \langle\Psi|\hat{B}|\psi_i\rangle\langle\psi_i|\Psi\rangle
= \sum_i \langle\psi_i|\Psi\rangle\langle\Psi|\hat{B}|\psi_i\rangle
= \sum_i \langle\psi_i|\hat{\rho}\hat{B}|\psi_i\rangle = \text{Tr}[\hat{\rho}\hat{B}]\]
for the trace of density operator
\[\text{Tr}[\hat{\rho}] = \sum_i \langle\psi_i|\hat{\rho}|\psi_i\rangle
= \sum_i \langle\psi_i|\Psi\rangle\langle\Psi|\psi_i\rangle
= \sum_i \langle\Psi|\psi_i\rangle\langle\psi_i|\Psi\rangle
= \langle\Psi|\Psi\rangle = 1\]
idempotent
\[\hat{\rho}^2 = (|\Psi\rangle\langle\Psi|)(|\Psi\rangle\langle\Psi|)
= |\Psi\rangle\langle\Psi| = \hat{\rho}\]
e.g. :
\[|\Psi\rangle = \frac{1}{\sqrt{2}}|\uparrow\rangle + \frac{1}{\sqrt{2}}|\downarrow\rangle
\]
\[\hat{\rho} = |\Psi\rangle\langle\Psi|
= \frac{1}{2}\left(|\uparrow\rangle\langle\uparrow| + |\uparrow\rangle\langle\downarrow|
+ |\downarrow\rangle\langle\uparrow| + |\downarrow\rangle\langle\downarrow|\right)
:= \frac{1}{2} \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\]
\(\heartsuit\) for a mixed state, we have the density operator
\[\hat{\rho} = \sum_j p_j|\Psi_j\rangle\langle\Psi_j|, \sum_jp_j = 1
\]
not idempotent
\[\begin{align*}
\hat{\rho}^2 &= \left(\sum_j p_j|\Psi_j\rangle\langle\Psi_j|\right)
\left(\sum_k p_k|\Psi_k\rangle\langle\Psi_k|\right) \\
&= \sum_{jk} p_jp_k|\Psi_j\rangle\langle\Psi_j|\Psi_k\rangle\langle\Psi_k| \\
&= \sum_j p_j^2|\Psi_j\rangle\langle\Psi_j| \neq \hat{\rho}
\end{align*}\]
e.g. :
\[\hat{\rho} = \frac{1}{2}(|\uparrow\rangle\langle\uparrow| + |\downarrow\rangle\langle\downarrow|)
:= \frac{1}{2} \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
2.3 Entangled state
direct product state \(\Rightarrow\) \(|\Psi_{AB}\rangle = |\Psi_A\rangle \otimes |\Psi_B\rangle\)
entangled state \(\Rightarrow\) \(\forall |\Psi_A\rangle, |\Psi_B\rangle\in\mathbb{R} \rightarrow |\Psi_{AB}\rangle \neq |\Psi_A\rangle \otimes |\Psi_B\rangle\)
e.g. :
Bell state
\[|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)
\]
\[|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)
\]
\[|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)
\]
\[|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)
\]
GHZ state
\[|GHZ\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)
\]
W state
\[|W\rangle = \frac{1}{\sqrt{3}}(|100\rangle + |010\rangle + |001\rangle)
\]
C3 - Second Quantization
3.1 - Identical particles
** permutation symmetry **
permutation symmetry \(\Rightarrow\) \([\hat{H}, \hat{P}_{ij}] = 0\), where \(\hat{P}_{ij}\) is the permutation operator
eigenvalue of permutation operator :
\[\hat{P}_{ij}^{-1} = \hat{P}_{ij} \Rightarrow \hat{P}_{ij}^2 = 1 \Rightarrow \lambda = \pm 1
\]
\[\begin{cases}
\lambda = +1 &\rightarrow \hat{P}_{ij}\psi^S = \psi^S
&\Rightarrow \text{Bosons }(s=0,\hbar,2\hbar,...) \\
\lambda = -1 &\rightarrow \hat{P}_{ij}\psi^A = -\psi^A
&\Rightarrow \text{Fermions }(s=\frac{\hbar}{2}, \frac{3\hbar}{2},...)
\end{cases}\]
** wavefunction **
for Fermions, the wavefunction follows the Pauli exclusion principle
\[\Psi_{k_1,...,k_N}(q_1,...,q_N) = \frac{1}{\sqrt{N!}}
\begin{bmatrix}
\phi_{k_1}(q_1) & \phi_{k_1}(q_2) & \dots & \phi_{k_1}(q_N) \\
\phi_{k_2}(q_1) & \phi_{k_2}(q_2) & \dots & \phi_{k_2}(q_N) \\
\vdots & \vdots & \ddots & \vdots\\
\phi_{k_N}(q_1) & \phi_{k_N}(q_2) & \dots & \phi_{k_N}(q_N) \\
\end{bmatrix}\]
for Bosons, particles are allowed to be in the same state
\[\Psi_{k_1,...,k_N}(q_1,...,q_N) = \sqrt{\frac{\prod_in_i!}{N!}} \sum_p\hat{P} [\phi_{k_1}(q_1) \cdots \phi_{k_1}(q_{n_1}) \phi_{k_2}(q_{n_1+1}) \cdots \phi_{k_2}(q_{n_1+n_2}) \cdots \phi_{k_N}(q_{N})]
\]
where \(n_i\) particles are in \(\phi_{k_i}\) state, and \(\hat{P}\) permutates particles in different states
3.2 - Particle number representation
identity of particles \(\Rightarrow\) we only need to distinguish between particles in different states
for Bosons, the wavefunction can be reformulated as
\[|\Psi\rangle := |n_1n_2...n_N\rangle
\]
where \(n_i\) denotes the particle number in the \(i\)th state
for Fermions, the wavefunction can be reformulated as
\[|\Psi\rangle := |k_1k_2...k_N\rangle
\]
where \(k_i = 0/1\) denotes the occupation of the \(i\)th state
** harmonoic oscillator (Bosons) **
Hamiltian of a one-dimensional harmonic oscillator
\[\hat{H} = \frac{1}{2m} \hat{p}_x^2 + \frac{1}{2} m\omega^2\hat{x}^2
\]
define creation and annihilation operator
\[\hat{a}^\dagger = \frac{1}{\sqrt{2}} \left(\sqrt{\frac{m\omega}{\hbar}}\hat{x} - i\frac{\hat{p}_x}{\sqrt{m\omega\hbar}}\right),
\hat{a} = \frac{1}{\sqrt{2}} \left(\sqrt{\frac{m\omega}{\hbar}}\hat{x} + i\frac{\hat{p}_x}{\sqrt{m\omega\hbar}}\right)\]
we have
\[[\hat{a}, \hat{a}^\dagger] = -\frac{i}{2\hbar} [\hat{x},\hat{p}_x] -\frac{i}{2\hbar} [\hat{x},\hat{p}_x] = 1
\]
\[\hat{x} = \sqrt{\frac{\hbar}{2m\omega}(\hat{a}^\dagger+\hat{a})},
\hat{p}_x = \sqrt{\frac{m\omega\hbar}{2}i(\hat{a}^\dagger-\hat{a})}\]
Hamiltian can be reformulated as
\[\hat{H} = (\hat{a}^\dagger\hat{a} + \hat{a}\hat{a}^\dagger) \frac{\hbar\omega}{2}
= (\hat{a}^\dagger\hat{a} + \frac{1}{2})\hbar\omega\]
if \(|n\rangle\) is an eigenstate of the Hamiltian with eigenvalue being \(E_n\), then we have
\[\hat{H}\hat{a}^\dagger|n\rangle
= \left(\hat{a}^\dagger\hat{a}\hat{a}^\dagger + \frac{1}{2}\hat{a}^\dagger\right) \hbar\omega|n\rangle
= \hat{a}^\dagger \left(\hat{a}^\dagger\hat{a} + \frac{1}{2} + 1\right) \hbar\omega|n\rangle
= (E_n + \hbar\omega) \hat{a}^\dagger |n\rangle\]
\[\hat{H}\hat{a}|n\rangle
= \left(\hat{a}^\dagger\hat{a}\hat{a} + \frac{1}{2}\hat{a}\right) \hbar\omega|n\rangle
= \hat{a} \left(\hat{a}^\dagger\hat{a} + \frac{1}{2} - 1\right) \hbar\omega|n\rangle
= (E_n - \hbar\omega) \hat{a} |n\rangle\]
which shows that \(\hat{a}^\dagger|n\rangle\) and \(\hat{a}|n\rangle\) will also be eigenstates of the Hamiltian
for the ground state \(|0\rangle\), we have
\[\hat{H}\hat{a}|0\rangle = (E_0 - \hbar\omega) \hat{a}|0\rangle
\]
because the eigenvalue can not be lowered any more, we have to make \(\hat{a}|0\rangle = 0\), which means
\[\hbar\omega\hat{a}^\dagger\hat{a}|0\rangle = \left(\hat{H}-\frac{1}{2}\hbar\omega\right)|0\rangle = 0
\Rightarrow \hat{H}|0\rangle = \frac{1}{2}\hbar\omega|0\rangle\]
eigenvalue of the ground state \(|0\rangle\) is \(E_0 = (1/2)\hbar\omega\)
then we can get all the eigenvalues and eigenstates of \(\hat{H}\)
\[E_n = \frac{1}{2}\hbar\omega + n\hbar\omega
\]
\[|n\rangle = \frac{1}{\sqrt{n!}} (\hat{a}^\dagger)^n |0\rangle
\]
** particle number operator **
creation and annihilation operator work as
\[\hat{a}^\dagger |n\rangle = \sqrt{n+1} |n+1\rangle
\]
\[\hat{a} |n\rangle = \sqrt{n} |n-1\rangle
\]
\[\langle n| \hat{a} = \sqrt{n+1} \langle n+1|
\]
\[\langle n| \hat{a}^\dagger = \sqrt{n} \langle n-1|
\]
define \(\hat{n} = \hat{a}^\dagger \hat{a}\), we have the commutation relations
\[[\hat{a}^\dagger, \hat{a}^\dagger] = 0, [\hat{a}, \hat{a}] = 0, [\hat{a}, \hat{a}^\dagger] = 1
\]
\[[\hat{a}, \hat{n}] = \hat{a}\hat{a}^\dagger\hat{a} - \hat{a}^\dagger\hat{a}\hat{a}
= (\hat{a}\hat{a}^\dagger - \hat{a}^\dagger\hat{a})\hat{a} = [\hat{a}, \hat{a}^\dagger]\hat{a}
= \hat{a}\]
\[[\hat{a}^\dagger, \hat{n}] = \hat{a}^\dagger\hat{a}^\dagger\hat{a} - \hat{a}^\dagger\hat{a}\hat{a}^\dagger = \hat{a}^\dagger(\hat{a}^\dagger\hat{a} - \hat{a}\hat{a}^\dagger)
= \hat{a}^\dagger[\hat{a}^\dagger, \hat{a}] = -\hat{a}^\dagger\]
** \(N\)-dimensional harmonic oscillator (Bosons) **
commutation relations
\[[\hat{a}_i^\dagger, \hat{a}_j^\dagger] = 0, [\hat{a}_i, \hat{a}_j] = 0,
[\hat{a}_i, \hat{a}_j^\dagger] = \delta_{ij}\]
eigenstates and eigenvalues
\[|n_1n_2...n_N\rangle = \frac{1}{\sqrt{\prod_in_i!}}
\left(\hat{a}_1^\dagger\right)^{n_1} \left(\hat{a}_2^\dagger\right)^{n_2} ...
\left(\hat{a}_N^\dagger\right)^{n_N} |0\rangle\]
\[E_{n_1n_2n_N} = \sum_{i=1}^N \left(n_i + \frac{1}{2}\right) \hbar\omega
\]
where \(n_i\) is the particle number in the \(i\)th state
creation and annihilation operators for Bosons behave
\[\hat{a}^\dagger_i|n_1n_2...n_i...\rangle = \sqrt{n_i+1}|n_1n_2...(n_i+1)...\rangle
\]
\[\hat{a}_i|n_1n_2...n_i...\rangle = \sqrt{n_i}|n_1n_2...(n_i-1)...\rangle
\]
\[\langle...n_i...n_2n_1|\hat{a}_i = \sqrt{n_i+1}\langle...(n_i+1)...n_2n_1|
\]
\[\langle...n_i...n_2n_1|\hat{a}^\dagger_i = \sqrt{n_i}\langle...(n_i-1)...n_2n_1|
\]
\[\hat{n}_i|n_1n_2...n_i...\rangle = \hat{a}^\dagger_i\hat{a}_i|n_1n_2...n_i...\rangle
= n_i|n_1n_2...n_i...\rangle\]
** Fermions **
for Fermions, basis of particle number representation is
\[|n_1n_2...n_N\rangle, n_\alpha=0,1
\]
where \(n_\alpha\) is the occupation number of the \(\alpha\)th state
creation and annihilation operators for Fermions behave
\[\hat{a}^\dagger_\alpha|n_1n_2...n_\alpha...\rangle = (-1)^{\sum_{\beta=1}^{\alpha-1}n_\beta} |n_1n_2...1_\alpha...\rangle \delta_{n_\alpha0}
\]
\[\hat{a}_\alpha|n_1n_2...n_\alpha...\rangle = (-1)^{\sum_{\beta=1}^{\alpha-1}n_\beta}|n_1n_2...0_\alpha...\rangle\delta_{n_\alpha1}
\]
\[\langle...n_\alpha...n_2n_1|\hat{a}_\alpha = (-1)^{\sum_{\beta=1}^{\alpha-1}n_\beta}\langle...1_\alpha...n_2n_1|\delta_{n_\alpha0}
\]
\[\langle...n_\alpha...n_2n_1|\hat{a}^\dagger_\alpha = (-1)^{\sum_{\beta=1}^{\alpha-1}n_\beta}\langle...0_\alpha...n_2n_1|\delta_{n_\alpha1}
\]
\[\hat{n}_\alpha|n_1n_2...n_\alpha...\rangle = \hat{a}^\dagger_\alpha\hat{a}_\alpha|n_1n_2...n_\alpha...\rangle = \delta_{n_\alpha1}|n_1n_2...1_\alpha...\rangle
\]
commutation relations
\[[\hat{a}_\alpha^\dagger, \hat{a}_\beta^\dagger]_+ = 0, [\hat{a}_\alpha, \hat{a}_\beta]_+ = 0,
[\hat{a}_\alpha, \hat{a}_\beta^\dagger]_+ = \delta_{\alpha\beta}\]
3.3 - Single and double particle operator of Bosons
** single-particle operator **
in the particle number representation, single particle operator becomes
\[\hat{F} = \sum_{ij}f_{ij}\hat{a}_i^\dagger\hat{a}_j, f_{ij} = \langle\psi_i|\hat{f}|\psi_j\rangle
\]
for diagonal element
\[\begin{align*}
\bar{\hat{F}} &= \langle...n_l...n_k...|\hat{F}|...n_k...n_l...\rangle \\
&= \sum_{ij}f_{ij} \langle...n_l...n_k...|\hat{a}_i^\dagger\hat{a}_j|...n_k...n_l...\rangle \\
&= \sum_{i}f_{ii} \langle...n_l...n_k...|\hat{a}_i^\dagger\hat{a}_i|...n_k...n_l...\rangle \\
&= \sum_{i}f_{ii} \langle...n_l...n_k...|\hat{n}_i|...n_k...n_l...\rangle \\
&= \sum_{i}f_{ii}n_i
\end{align*}\]
for single-excitation element
\[\begin{align*}
& \langle...(n_l-1)...(n_k+1)...|\hat{F}|...n_k...n_l...\rangle \\
=& \sum_{ij}f_{ij} \langle...(n_l-1)...(n_k+1)...|\hat{a}_i^\dagger\hat{a}_j|...n_k...n_l...\rangle \\
=& \sum_{i}f_{ii} \sqrt{n_k+1}\delta_{ik} \langle...(n_l-1)...n_k...|...n_k...(n_l-1)...\rangle \sqrt{n_l}\delta_{jl} \\
=& f_{kl} \sqrt{(n_k+1)n_l}
\end{align*}\]
** double-particle operator **
in the particle number representation, double particle operator becomes
\[\hat{G} = \frac{1}{2}\sum_{i'j'ij}g_{i'j'ji}\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}\hat{a}_j\hat{a}_i,
g_{i'j'ji} = \langle\psi_{i'}(1)\psi_{j'}(2)|\hat{g}(1,2)|\psi_j(2)\psi_i(1)\rangle\]
from the symmetry \(\hat{g}(a,b) = \hat{g}(b,a)\), we have
\[g_{i'j'ji} = \langle\psi_{i'}(1)\psi_{j'}(2)|\hat{g}(1,2)|\psi_j(2)\psi_i(1)\rangle
= \langle\psi_{i'}(2)\psi_{j'}(1)|\hat{g}(2,1)|\psi_j(1)\psi_i(2)\rangle = g_{j'i'ij}\]
for diagonal element
\[\begin{align*}
\langle...n_j...n_i...|\hat{G}|...n_i...n_j...\rangle =& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...n_j...n_i...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}\hat{a}_j\hat{a}_i|...n_i...n_j...\rangle \\
+& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...n_i...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}\hat{a}_j\hat{a}_i|...n_i...\rangle
\end{align*}\]
\[\begin{align*}
& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...n_j...n_i...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}
\hat{a}_j\hat{a}_i|...n_i...n_j...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...n_j...n_i...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}
|...(n_i-1)...(n_j-1)...\rangle \sqrt{n_in_j} \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...(n_j-1)...(n_i-1)...|...(n_i-1)...(n_j-1)...\rangle (\delta_{ii'}\delta_{jj'} + \delta_{ij'}\delta_{ji'}) n_in_j \\
=& \frac{1}{2} \sum_{ij} (g_{ijji} + g_{jiji}) n_in_j
\end{align*}\]
\[\begin{align*}
& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...n_i...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}\hat{a}_j\hat{a}_i|...n_i...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...n_i...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}
|...(n_i-2)...\rangle \delta_{ij} \sqrt{n_i(n_i-1)} \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...(n_i-2)...|...(n_i-2)...\rangle \delta_{ii'}\delta_{ij'} \delta_{ij} n_i(n_i-1) \\
=& \frac{1}{2} \sum_{i} n_i(n_i-1) g_{iiii}
\end{align*}\]
\[\bar{\hat{G}} = \frac{1}{2}\sum_{ij}n_in_j(g_{ijji}+g_{jiji}) + \frac{1}{2}\sum_{i}n_i(n_i-1)g_{iiii}
\]
for double excitation \((k,l)\rightarrow(a,b)\)
\[\begin{align*}
& \langle...(n_b+1)...(n_a+1)...(n_l-1)...(n_k-1)...|\hat{G}|...n_k...n_l...n_a...n_b...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...(n_b+1)...(n_a+1)...(n_l-1)...(n_k-1)...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}\hat{a}_j\hat{a}_i|...n_k...n_l...n_a...n_b...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \sqrt{(n_b+1)(n_a+1)}\sqrt{n_kn_l} (\delta_{i'b}\delta_{j'a} + \delta_{i'a}\delta_{j'b}) (\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) \\
& \langle...n_b...n_a...(n_l-1)...(n_k-1)...|...(n_k-1)...(n_l-1)...n_a...n_b...\rangle \\
=& \frac{1}{2} \sum_{ij} \sqrt{(n_b+1)(n_a+1)n_kn_l} (g_{baji} + g_{abji}) (\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) \\
=& \frac{1}{2} \sqrt{(n_b+1)(n_a+1)n_kn_l} (g_{balk} + g_{ablk} + g_{bakl} + g_{abkl}) \\
=& (g_{ablk} + g_{abkl}) \sqrt{(n_b+1)(n_a+1)n_kn_l}
\end{align*}\]
for double excitation \((k,l)\rightarrow(a,a)\)
\[\begin{align*}
& \langle...(n_a+2)...(n_l-1)...(n_k-1)...|\hat{G}|...n_k...n_l...n_a...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...(n_a+2)...(n_l-1)...(n_k-1)...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}\hat{a}_j\hat{a}_i|...n_k...n_l...n_a...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \sqrt{(n_a+2)(n_a+1)}\sqrt{n_kn_l} \cdot \delta_{i'a}\delta_{j'a} (\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) \\
& \langle...n_a...(n_l-1)...(n_k-1)...|...(n_k-1)...(n_l-1)...n_a...\rangle \\
=& \frac{1}{2} \sum_{ij} \sqrt{(n_a+2)(n_a+1)n_kn_l} \cdot g_{aaji}(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) \\
=& \frac{1}{2} \sqrt{(n_a+2)(n_a+1)n_kn_l} (g_{aalk} + g_{aakl}) \\
=& g_{aalk} \sqrt{(n_a+2)(n_a+1)n_kn_l}
\end{align*}\]
for double excitation \((k,k)\rightarrow(a,b)\)
\[\begin{align*}
& \langle...(n_b+1)...(n_a+1)...(n_k-2)...|\hat{G}|...n_k...n_a...n_b...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...(n_b+1)...(n_a+1)...(n_k-2)...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}\hat{a}_j\hat{a}_i|...n_k...n_a...n_b...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \sqrt{(n_b+1)(n_a+1)n_k(n_k-1)} \cdot (\delta_{i'b}\delta_{j'a} + \delta_{i'a}\delta_{j'b}) \delta_{ik}\delta_{jk} \\
& \langle...n_b...n_a...(n_k-2)...|...(n_k-2)...n_a...n_b...\rangle \\
=& \frac{1}{2} \sum_{ij} \sqrt{(n_b+1)(n_a+1)n_k(n_k-1)} \cdot (g_{baji} + g_{abji}) \delta_{ik}\delta_{jk} \\
=& \frac{1}{2} \sqrt{(n_b+1)(n_a+1)n_k(n_k-1)} \cdot (g_{bakk} + g_{abkk}) \\
=& g_{abkk} \sqrt{(n_b+1)(n_a+1)n_k(n_k-1)}
\end{align*}\]
for double excitation \((k,k)\rightarrow(a,a)\)
\[\begin{align*}
& \langle...(n_a+2)...(n_k-2)...|\hat{G}|...n_k...n_a...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \langle...(n_a+2)...(n_k-2)...|\hat{a}^\dagger_{i'}\hat{a}^\dagger_{j'}\hat{a}_j\hat{a}_i|...n_k...n_a...\rangle \\
=& \frac{1}{2} \sum_{i'j'ij}g_{i'j'ji} \sqrt{(n_a+2)(n_a+1)n_k(n_k-1)} \cdot \delta_{i'a}\delta_{j'a} \delta_{ik}\delta_{jk} \\
& \langle...n_a...(n_k-2)...|...(n_k-2)...n_a...\rangle \\
=& \frac{1}{2} \sqrt{(n_a+2)(n_a+1)n_k(n_k-1)} \cdot g_{aakk}
\end{align*}\]
3.4 - Single and double particle operator of Fermions
** single-particle operator **
for diagonal element
\[\begin{align*}
\bar{\hat{F}} &= \langle...n_\alpha...|\hat{F}|...n_\alpha...\rangle \\
&= \sum_{\alpha\beta}f_{\alpha\beta} \langle...n_\alpha...|\hat{a}_\alpha^\dagger\hat{a}_\beta|...n_\alpha...\rangle \\
&= \sum_{\alpha}f_{\alpha\alpha} \langle...n_\alpha...|\hat{a}_\alpha^\dagger\hat{a}_\alpha|...n_\alpha...\rangle \\
&= \sum_{\alpha}f_{\alpha\alpha} \langle...n_\alpha...|\hat{n}_\alpha|...n_\alpha...\rangle \\
&= \sum_{\alpha}f_{\alpha\alpha} \delta_{n_\alpha1}
\end{align*}\]
for single-excitation element
\[\begin{align*}
& \langle...0_\gamma...1_a...|\hat{F}|...0_a...1_\gamma...\rangle \text{ }(\gamma > a) \\
=& \sum_{\alpha\beta}f_{\alpha\beta} \langle...0_\gamma...1_a...|\hat{a}_\alpha^\dagger\hat{a}_\beta|...0_a...1_\gamma...\rangle \\
=& \sum_{\alpha\beta}f_{\alpha\beta} (-1)^{\sum_{i=1}^{a-1}n_i} (-1)^{\sum_{i=1}^{\gamma-1}n_i} \delta_{a\alpha}\delta_{\gamma\beta} \langle...0_\gamma...0_a...|...0_a...0_\gamma...\rangle \\
=& f_{a\gamma} (-1)^{\sum_{i=a+1}^{\gamma-1}n_i}
\end{align*}\]
** Wick's theorem **
define the Fock vacuum which consists of \(N\) fermions
\[|\rangle = |ijk...\rangle = \prod_{i\leq N} \hat{a}_i^\dagger |0\rangle
\]
from the relations \(\hat{a}_a |\rangle = 0\) and \(\hat{a}_i^\dagger |\rangle = 0\), we can define the quasi-particle operator
\[\hat{\alpha}_a^\dagger = \hat{a}_a^\dagger \qquad \hat{\alpha}_a = \hat{a}_a \\
\hat{\alpha}_i^\dagger = \hat{a}_i \qquad \hat{\alpha}_i = \hat{a}_i^\dagger\]
where we have the relation \(\hat{\alpha}_\nu |\rangle = 0\)
in the normal product, the creation operators of quasi-particle should be in the left of the annihilation operators, e.g. :
\[\{\hat{a}_a^\dagger\hat{a}_i^\dagger\hat{a}_j\hat{a}_b\} = -\hat{a}_a^\dagger\hat{a}_j\hat{a}_i^\dagger\hat{a}_b
\]
the diagonal element of normal product for Fock vacuum should be zero
\[\langle|\{AB...\}|\rangle = \langle|...\hat{\alpha}_\nu|\rangle = 0
\]
define contraction as the diagonal element of the operator
\[\acute{A}\grave{B} = \langle|AB|\rangle
\]
note that \acute and \grave are used for Wick contraction due to Markdown environment
we can notice that contraction should be made between creation and annihilation operators
\[\acute{\hat{a}_\mu^\dagger}\grave{\hat{a}_\nu^\dagger} = \langle|\hat{a}_\mu^\dagger\hat{a}_\nu^\dagger|\rangle = 0 \qquad \acute{\hat{a}_\mu}\grave{\hat{a}_\nu} = \langle|\hat{a}_\mu\hat{a}_\nu|\rangle = 0
\]
the Wick's theorem goes
\[\begin{align*}
ABCD... &= \{ABCD...\} &\text{no contraction} \\
&+ \{\acute{A}\grave{B}CD...\} + \{A\acute{B}\grave{C}D...\} + ... &\text{one contraction} \\
&+ \{\acute{A}\grave{B}\acute{C}\grave{D}...\} + ... &\text{two contraction} \\
&+ ... + \{\acute{A}\grave{B}\acute{C}\grave{D}\acute{E}\grave{F}...\} &\text{more contractions}
\end{align*}\]
for two-body interactions
\[\begin{align*}
\hat{a}_{\alpha'}^\dagger\hat{a}_{\beta'}^\dagger\hat{a}_\beta\hat{a}_\alpha
&= \langle|\hat{a}_{\alpha'}^\dagger\hat{a}_\alpha|\rangle \langle|\hat{a}_{\beta'}^\dagger\hat{a}_\beta|\rangle - \langle|\hat{a}_{\alpha'}^\dagger\hat{a}_\beta|\rangle \langle|\hat{a}_{\beta'}^\dagger\hat{a}_\alpha|\rangle \\
&+ \langle|\hat{a}_{\alpha'}^\dagger\hat{a}_\alpha|\rangle \{\hat{a}_{\beta'}^\dagger\hat{a}_\beta\} + \langle|\hat{a}_{\beta'}^\dagger\hat{a}_\beta|\rangle \{\hat{a}_{\alpha'}^\dagger\hat{a}_\alpha\} \\
&- \langle|\hat{a}_{\alpha'}^\dagger\hat{a}_\beta|\rangle \{\hat{a}_{\beta'}^\dagger\hat{a}_\alpha\} - \langle|\hat{a}_{\beta'}^\dagger\hat{a}_\alpha|\rangle \{\hat{a}_{\alpha'}^\dagger\hat{a}_\beta\} \\
&+ \{\hat{a}_{\alpha'}^\dagger\hat{a}_{\beta'}^\dagger\hat{a}_\beta\hat{a}_\alpha\}
\end{align*}\]
** double-particle operator **
\[\begin{align*}
\langle|\hat{G}|\rangle &= \frac{1}{2}\sum_{\alpha\beta\alpha'\beta'}g_{\alpha'\beta'\beta\alpha}\langle|\hat{a}_{\alpha'}^\dagger\hat{a}_{\beta'}^\dagger\hat{a}_\beta\hat{a}_\alpha|\rangle \\
&= \frac{1}{2}\sum_{\alpha\beta\alpha'\beta'}g_{\alpha'\beta'\beta\alpha} \left(\langle|\hat{a}_{\alpha'}^\dagger\hat{a}_\alpha|\rangle \langle|\hat{a}_{\beta'}^\dagger\hat{a}_\beta|\rangle - \langle|\hat{a}_{\alpha'}^\dagger\hat{a}_\beta|\rangle \langle|\hat{a}_{\beta'}^\dagger\hat{a}_\alpha|\rangle\right) \\
&= \frac{1}{2}\sum_{\alpha \neq \beta} n_\alpha n_\beta (g_{\alpha\beta\beta\alpha} - g_{\beta\alpha\beta\alpha})
\end{align*}\]
高等量子力学学习笔记，Hilbert空间，二次量子化
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