Week 2: Second Quantization and Collective Excitations National Tsing Hua University
August 1, 2026
Reference: Blaizot & Ripka (1986)
Central Pedagogical Goal ¶ The main goal of this lecture is to understand second quantization not as a new physical theory, but as a more natural language for describing quantum many-body systems , especially when particle number, occupation numbers, interactions, and collective excitations become the relevant degrees of freedom.
A useful guiding question throughout the lecture is:
What are the appropriate variables for describing a quantum many-body system?
The conceptual chain of this lecture is
single-particle basis → many-particle basis → occupation numbers → a † , a → second-quantized Hamiltonian → collective excitations \boxed{
\text{single-particle basis}
\rightarrow
\text{many-particle basis}
\rightarrow
\text{occupation numbers}
\rightarrow
a^\dagger,a
\rightarrow
\text{second-quantized Hamiltonian}
\rightarrow
\text{collective excitations}
} single-particle basis → many-particle basis → occupation numbers → a † , a → second-quantized Hamiltonian → collective excitations 1. Why Do We Need Second Quantization? ¶ 1.1 Occupation-number representation ¶ It doesn’t make sense to keep track of each particle when particles are indistinguishable ¶ Consider a set of single-particle states
∣ ϕ 1 ⟩ , ∣ ϕ 2 ⟩ , … , ∣ ϕ M ⟩ . |\phi_1\rangle,\,
|\phi_2\rangle,\,
\ldots,\,
|\phi_M\rangle. ∣ ϕ 1 ⟩ , ∣ ϕ 2 ⟩ , … , ∣ ϕ M ⟩ . For two distinguishable particles, the Hilbert space is
H 1 ⊗ H 1 , \mathcal H_1\otimes \mathcal H_1, H 1 ⊗ H 1 , and a basis can be written as
∣ ϕ i ⟩ ⊗ ∣ ϕ j ⟩ . |\phi_i\rangle\otimes|\phi_j\rangle. ∣ ϕ i ⟩ ⊗ ∣ ϕ j ⟩ . However, identical quantum particles are indistinguishable .
For two identical bosons, the properly symmetrized state is
∣ Ψ i j ( B ) ⟩ = 1 2 ( ∣ i ⟩ ∣ j ⟩ + ∣ j ⟩ ∣ i ⟩ ) , |\Psi_{ij}^{(B)}\rangle
=
\frac{1}{\sqrt{2}}
\left(
|i\rangle|j\rangle
+
|j\rangle|i\rangle
\right), ∣ Ψ ij ( B ) ⟩ = 2 1 ( ∣ i ⟩ ∣ j ⟩ + ∣ j ⟩ ∣ i ⟩ ) , while for two identical fermions,
∣ Ψ i j ( F ) ⟩ = 1 2 ( ∣ i ⟩ ∣ j ⟩ − ∣ j ⟩ ∣ i ⟩ ) . |\Psi_{ij}^{(F)}\rangle
=
\frac{1}{\sqrt{2}}
\left(
|i\rangle|j\rangle
-
|j\rangle|i\rangle
\right). ∣ Ψ ij ( F ) ⟩ = 2 1 ( ∣ i ⟩ ∣ j ⟩ − ∣ j ⟩ ∣ i ⟩ ) . For many particles, explicitly symmetrizing or antisymmetrizing wavefunctions quickly becomes cumbersome.
Instead of asking
Which particle occupies which state?
it is more natural to ask
How many particles occupy each state?
This motivates the occupation-number representation for an N N N
A many-particle state can be written as
∣ n α 1 , n α 2 , … , n α M ⟩ , |n_{\alpha_1},n_{\alpha_2},\ldots,n_{\alpha_M}\rangle, ∣ n α 1 , n α 2 , … , n α M ⟩ , where n α j n_{\alpha_j} n α j α j \alpha_j α j
For bosons,
n α j = 0 , 1 , 2 , … , n_{\alpha_j}=0,1,2,\ldots, n α j = 0 , 1 , 2 , … , while for fermions,
n α j = 0 , 1. n_{\alpha_j}=0,1. n α j = 0 , 1. Thus, instead of tracking individual particle labels, we describe the physical configuration directly by occupation numbers.
A useful conceptual shorthand is:
First quantization emphasizes particle coordinates.
1.2 The exponential growing Hilbert space dimension ¶ Dimension of many-body Hilbert space grows exponentially, but local Hamiltonians usually is specified by a small set of parameters ¶ When we have N N N
H N = H 1 ( 1 ) ⊗ H 1 ( 2 ) ⊗ . . . ⊗ H 1 ( N ) . \mathcal{H}_N=\mathcal{H}_1^{(1)}\otimes\mathcal{H}_1^{(2)}\otimes...\otimes\mathcal{H}_1^{(N)}. H N = H 1 ( 1 ) ⊗ H 1 ( 2 ) ⊗ ... ⊗ H 1 ( N ) . If D [ H 1 ] = d D[\mathcal{H}_1]=d D [ H 1 ] = d d d d H 1 \mathcal{H}_1 H 1 D [ H N ] ∼ e N D[\mathcal{H}_N]\sim e^N D [ H N ] ∼ e N
It means the matrix representation of many-body Hamiltonians(local) are usually sparse. How to effectively compress the information and form a informative description is the key.
Faithful representations: first quantization, second quantization(suitable for later simplification focusing on collective behavior)
Variational representations: tensor networks, Boltzmann machine, etc.
2. N N N ¶ 2.1 From the first quantization to the occupation number representation ¶ Let’s denote the real space coordinate as r ⃗ \vec{r} r σ \sigma σ x x x i.e. ∣ x ⟩ ≡ ∣ r ⃗ , σ ⟩ |x\rangle \equiv|\vec{r},\sigma\rangle ∣ x ⟩ ≡ ∣ r , σ ⟩
The completeness relation is
∫ d x ∣ x ⟩ ⟨ x ∣ = 1 r ⃗ , σ ; ⟨ x ∣ x ′ ⟩ = δ ( x − x ′ ) . \int dx |x\rangle\langle x|=\mathbf{1}_{\vec{r},\sigma}; \langle x|x'\rangle=\delta(x-x'). ∫ d x ∣ x ⟩ ⟨ x ∣ = 1 r , σ ; ⟨ x ∣ x ′ ⟩ = δ ( x − x ′ ) . Here, ∫ d x . . . \int dx ... ∫ d x ... ∫ d r ⃗ ∑ σ . . . \int d\vec{r}\sum_{\sigma}... ∫ d r ∑ σ ... δ ( x − x ′ ) \delta(x-x') δ ( x − x ′ ) δ ( r ⃗ − r ′ ⃗ ) δ σ , σ ′ \delta(\vec{r}-\vec{r'})\delta_{\sigma,\sigma'} δ ( r − r ′ ) δ σ , σ ′
The wave function, φ α ( x ) = ⟨ x ∣ α ⟩ \varphi_{\alpha}(x)=\langle x|\alpha\rangle φ α ( x ) = ⟨ x ∣ α ⟩ x x x ∣ α ⟩ |\alpha\rangle ∣ α ⟩
Let’s consider the N N N { α i } \{\alpha_i\} { α i } ∣ { α i } ) |\{\alpha_i\}) ∣ { α i })
∣ α 1 , α 2 , . . . , α N ) ≡ ∣ α 1 ⟩ ∣ α 2 ⟩ . . . ∣ α N ⟩ |\alpha_1,\alpha_2,...,\alpha_N)\equiv|\alpha_1\rangle|\alpha_2\rangle...|\alpha_N\rangle ∣ α 1 , α 2 , ... , α N ) ≡ ∣ α 1 ⟩ ∣ α 2 ⟩ ...∣ α N ⟩ Here, we assume the compelteness relation in H 1 \mathcal{H}_1 H 1 ∑ α ∣ α ⟩ ⟨ α ∣ = 1 H 1 \sum_{\alpha}|\alpha\rangle\langle\alpha|=\textbf{1}_{\mathcal{H}_1} ∑ α ∣ α ⟩ ⟨ α ∣ = 1 H 1 ∣ . . . ) |...) ∣... ) NOT the physical wave function of N N N N N N
To generate wave functions for indistinguishable particles, we consider the symmetrize and anti-symmetrize operators.
S = ( N ! ) − 1 ∑ P P A = ( N ! ) − 1 ∑ P ( − 1 ) P P \begin{aligned}
S &= (N!)^{-1}\sum_P P \\
A &= (N!)^{-1}\sum_P (-1)^P P
\end{aligned} S A = ( N ! ) − 1 P ∑ P = ( N ! ) − 1 P ∑ ( − 1 ) P P where P P P ( − 1 ) P (-1)^P ( − 1 ) P P P P P ∗ P^* P ∗ P ∗ ( A B C ) = ( C A B ) P^*(ABC)=(CAB) P ∗ ( A BC ) = ( C A B ) P ∗ P^* P ∗ ( C A B ) (CAB) ( C A B ) ( A B C ) (ABC) ( A BC ) C A CA C A C B CB CB TWO nearest neighbor swaps.
Now, let’s apply the symmetrization and anti-symmetrization operators to our N N N ∣ . . . ) |...) ∣... )
2.2 Bosons ¶ ∣ α 1 α 2 . . . α N ⟩ S = N S S ∣ α 1 α 2 . . . α N ) . |\alpha_1\alpha_2...\alpha_N\rangle_S= N_S S|\alpha_1\alpha_2...\alpha_N). ∣ α 1 α 2 ... α N ⟩ S = N S S ∣ α 1 α 2 ... α N ) . Here N S N_S N S S ⟨ α 1 α 2 . . . α N ∣ α 1 α 2 . . . α N ⟩ S = 1 _S\langle \alpha_1\alpha_2...\alpha_N|\alpha_1\alpha_2...\alpha_N\rangle_S=1 S ⟨ α 1 α 2 ... α N ∣ α 1 α 2 ... α N ⟩ S = 1 ⟨ α ∣ β ⟩ = δ α , β \langle \alpha|\beta\rangle=\delta_{\alpha,\beta} ⟨ α ∣ β ⟩ = δ α , β ∣ { α i } ) |\{\alpha_i\}) ∣ { α i }) α \alpha α
It is the time that the occupation number representation becomes useful. Instead of tracking the quantum number of individual particle, which does not make sense for identicle particles, we describe the system as how many particles occupy a particular quantum state ∣ α ⟩ |\alpha\rangle ∣ α ⟩ H 1 \mathcal{H}_1 H 1
That is, if we have a M M M H 1 \mathcal{H}_1 H 1 a j ; j = 1 ∼ M a_j;j=1\sim M a j ; j = 1 ∼ M ∣ { α i } ) |\{\alpha_i\}) ∣ { α i }) α i ∈ { a j } \alpha_i\in\{a_j\} α i ∈ { a j } ∣ n a 1 n a 2 . . . n a M ⟩ |n_{a_1}n_{a_2}...n_{a_M}\rangle ∣ n a 1 n a 2 ... n a M ⟩ ∑ j n j = N \sum_jn_j=N ∑ j n j = N M M M ∣ n a 1 n a 2 . . . ⟩ |n_{a_1}n_{a_2}...\rangle ∣ n a 1 n a 2 ... ⟩ last single particle state in H 1 \mathcal{H}_1 H 1
With the occupation number notion in mind, we know the counting problem better. For a fictitious state ∣ α 1 α 2 . . . α N ) |\alpha_1\alpha_2...\alpha_N) ∣ α 1 α 2 ... α N ) ∣ n a 1 n a 2 . . . ⟩ |n_{a_1}n_{a_2}...\rangle ∣ n a 1 n a 2 ... ⟩ ∑ j n a j = N \sum_j n_{a_j}=N ∑ j n a j = N
N ! n a 1 ! n a 2 ! . . . . \frac{N!}{n_{a_1}!n_{a_2}!...}. n a 1 ! n a 2 ! ... N ! . Therefore, we can find N s N_s N s
S ⟨ α 1 α 2 . . . α N ∣ α 1 α 2 . . . α N ⟩ S = ⟨ n α 1 n α 2 . . . ∣ n α 1 n α 2 . . . ⟩ = 1 = N S 2 ∑ P ∑ Q ( N ! ) − 2 ( α 1 α 2 . . . α N ∣ P Q ∣ α 1 α 2 . . . α N ) \begin{aligned}
_S\langle \alpha_1\alpha_2...\alpha_N|\alpha_1\alpha_2...\alpha_N\rangle_S&=\langle n_{\alpha_1}n_{\alpha_2}...|n_{\alpha_1}n_{\alpha_2}...\rangle=1\\
&=N_S^2\sum_P\sum_Q (N!)^{-2}(\alpha_1\alpha_2...\alpha_N|PQ|\alpha_1\alpha_2...\alpha_N)
\end{aligned} S ⟨ α 1 α 2 ... α N ∣ α 1 α 2 ... α N ⟩ S = ⟨ n α 1 n α 2 ...∣ n α 1 n α 2 ... ⟩ = 1 = N S 2 P ∑ Q ∑ ( N ! ) − 2 ( α 1 α 2 ... α N ∣ PQ ∣ α 1 α 2 ... α N ) ( α 1 α 2 . . . α N ∣ P = ( α P 1 α P 2 . . . α P N ∣ (\alpha_1\alpha_2...\alpha_N|P=(\alpha_{P_1}\alpha_{P_2}...\alpha_{P_N}| ( α 1 α 2 ... α N ∣ P = ( α P 1 α P 2 ... α P N ∣ { α i } \{\alpha_i\} { α i } Q ∣ α 1 α 2 . . . α N ) = ∣ α Q 1 α Q 2 . . . α Q N ) Q|\alpha_1\alpha_2...\alpha_N)=|\alpha_{Q_1}\alpha_{Q_2}...\alpha_{Q_N}) Q ∣ α 1 α 2 ... α N ) = ∣ α Q 1 α Q 2 ... α Q N ) P P P Q Q Q P P P Q Q Q n α 1 ! n α 2 ! . . . n_{\alpha_1}!n_{\alpha_2}!... n α 1 ! n α 2 ! ... Q Q Q
1 = N S 2 ∑ P ( N ! ) − 2 n α 1 ! n α 2 ! . . . = N S 2 ( N ! ) − 1 n α 1 ! n α 2 ! . . . 1=N_S ^2\sum_P(N!)^{-2}n_{\alpha_1}!n_{\alpha_2}!...=N_S^2(N!)^{-1} n_{\alpha_1}!n_{\alpha_2}!... 1 = N S 2 P ∑ ( N ! ) − 2 n α 1 ! n α 2 ! ... = N S 2 ( N ! ) − 1 n α 1 ! n α 2 ! ... The summation over P P P N ! N! N ! P P P
This gives
N S = N ! n α 1 ! n α 2 ! . . . N_S=\sqrt{\frac{N!}{n_{\alpha_1}!n_{\alpha_2}!...}} N S = n α 1 ! n α 2 ! ... N ! and
∣ α 1 α 2 . . . α N ⟩ S = N ! n α 1 ! n α 2 ! . . . S ∣ α 1 α 2 . . . α N ) = 1 N ! n α 1 ! n α 2 ! . . . ∑ P ∣ α P 1 α P 2 . . . α N ) = ∣ n α 1 n α 2 . . . ⟩ S \begin{aligned}
|\alpha_1\alpha_2...\alpha_N\rangle_S&=\sqrt{\frac{N!}{n_{\alpha_1}!n_{\alpha_2}!...}}S|\alpha_1\alpha_2...\alpha_N)\\
&=\frac{1}{\sqrt{N!n_{\alpha_1}!n_{\alpha_2}!...}}\sum_P |\alpha_{P_1}\alpha_{P_2}...\alpha_N)\\
&=|n_{\alpha_1}n_{\alpha_2}...\rangle_S
\end{aligned} ∣ α 1 α 2 ... α N ⟩ S = n α 1 ! n α 2 ! ... N ! S ∣ α 1 α 2 ... α N ) = N ! n α 1 ! n α 2 ! ... 1 P ∑ ∣ α P 1 α P 2 ... α N ) = ∣ n α 1 n α 2 ... ⟩ S The quantum number α \alpha α α ′ \alpha' α ′
S ⟨ { α i } ∣ { α i ′ } ⟩ S = 1 n α 1 ! . . . n α 1 ′ ! . . . ∑ P ⟨ α 1 ∣ α P 1 ′ ⟩ ⟨ α 2 ∣ α P 2 ′ ⟩ . . . ⟨ α N ∣ α P N ′ ⟩ ⏟ p e r m a n e n t . _S\langle \{\alpha_i\}|\{\alpha_i'\}\rangle_S=\frac{1}{\sqrt{n_{\alpha_1}!...n_{\alpha_1'}!...}} \underbrace{\sum_P\langle \alpha_1|\alpha_{P_1}'\rangle\langle \alpha_2|\alpha_{P_2}'\rangle...\langle \alpha_N|\alpha_{P_N}'\rangle}_{permanent}. S ⟨{ α i } ∣ { α i ′ } ⟩ S = n α 1 ! ... n α 1 ′ ! ... 1 p er man e n t P ∑ ⟨ α 1 ∣ α P 1 ′ ⟩ ⟨ α 2 ∣ α P 2 ′ ⟩ ... ⟨ α N ∣ α P N ′ ⟩ . S ⟨ { α i } ∣ { α i ′ } ⟩ S = N ! n α 1 ! . . . N ! n α 1 ′ ! . . . ( N ! ) − 2 ∑ P ∑ Q ( α 1 . . . ∣ P Q ∣ α 1 ′ . . . ) = N ! n α 1 ! . . . N ! n α 1 ′ ! . . . ( N ! ) − 2 ∑ P ∑ P Q ( α 1 . . . ∣ P Q ∣ α 1 ′ . . . ) = 1 n α 1 ! . . . n α 1 ′ ! . . . ∑ P Q ( α 1 . . . ∣ P Q ∣ α 1 ′ . . . ) = 1 n α 1 ! . . . n α 1 ′ ! . . . ∑ P ⟨ α 1 ∣ α P 1 ′ ⟩ ⟨ α 2 ∣ α P 2 ′ ⟩ . . . ⟨ α N ∣ α P N ′ ⟩ ⏟ p e r m a n e n t . \begin{aligned}
_S\langle \{\alpha_i\}|\{\alpha_i'\}\rangle_S&=\sqrt{\frac{N!}{n_{\alpha_1}!...}\frac{N!}{n_{\alpha_1'}!...}} (N!)^{-2}\sum_P\sum_Q (\alpha_1...|PQ|\alpha_1'...)\\
&=\sqrt{\frac{N!}{n_{\alpha_1}!...}\frac{N!}{n_{\alpha_1'}!...}} (N!)^{-2}\sum_P\sum_{PQ} (\alpha_1...|PQ|\alpha_1'...)\\
&=\frac{1}{\sqrt{n_{\alpha_1}!...n_{\alpha_1'}!...}} \sum_{PQ} (\alpha_1...|PQ|\alpha_1'...)\\
&=\frac{1}{\sqrt{n_{\alpha_1}!...n_{\alpha_1'}!...}} \underbrace{\sum_P\langle \alpha_1|\alpha_{P_1}'\rangle\langle \alpha_2|\alpha_{P_2}'\rangle...\langle \alpha_N|\alpha_{P_N}'\rangle}_{permanent}.
\end{aligned} S ⟨{ α i } ∣ { α i ′ } ⟩ S = n α 1 ! ... N ! n α 1 ′ ! ... N ! ( N ! ) − 2 P ∑ Q ∑ ( α 1 ...∣ PQ ∣ α 1 ′ ... ) = n α 1 ! ... N ! n α 1 ′ ! ... N ! ( N ! ) − 2 P ∑ PQ ∑ ( α 1 ...∣ PQ ∣ α 1 ′ ... ) = n α 1 ! ... n α 1 ′ ! ... 1 PQ ∑ ( α 1 ...∣ PQ ∣ α 1 ′ ... ) = n α 1 ! ... n α 1 ′ ! ... 1 p er man e n t P ∑ ⟨ α 1 ∣ α P 1 ′ ⟩ ⟨ α 2 ∣ α P 2 ′ ⟩ ... ⟨ α N ∣ α P N ′ ⟩ . Here, we have use the cyclic nature of permutation, so ∑ Q . . . P Q . . . = ∑ P Q . . . P Q . . . \sum_Q ...PQ...=\sum_{PQ}...PQ... ∑ Q ... PQ ... = ∑ PQ ... PQ ... ∑ P = N ! \sum_P=N! ∑ P = N !
2.3 Fermions ¶ ∣ α 1 α 2 . . . α N ⟩ A = N A A ∣ α 1 α 2 . . . α N ) . |\alpha_1\alpha_2...\alpha_N\rangle_A= N_A A|\alpha_1\alpha_2...\alpha_N). ∣ α 1 α 2 ... α N ⟩ A = N A A ∣ α 1 α 2 ... α N ) . We can perform exactly the same procedure and got N A N_{A} N A n α j = 0 , 1 ; n α j ! = 1 n_{\alpha_j}=0,1;n_{\alpha_j}!=1 n α j = 0 , 1 ; n α j ! = 1
N A = N ! N_A=\sqrt{N!} N A = N ! and
∣ α 1 α 2 . . . α N ⟩ A = N ! A ∣ α 1 α 2 . . . α N ) = 1 N ! ∑ P ( − 1 ) P ∣ α P 1 α P 2 . . . α P N ) = ∣ n α 1 n α 2 . . . ⟩ A . \begin{aligned}
|\alpha_1\alpha_2...\alpha_N\rangle_A&=\sqrt{N!}A|\alpha_1\alpha_2...\alpha_N)\\
&=\frac{1}{\sqrt{N!}}\sum_P (-1)^P|\alpha_{P_1}\alpha_{P_2}...\alpha_{P_N})\\
&=|n_{\alpha_1}n_{\alpha_2}...\rangle_A.
\end{aligned} ∣ α 1 α 2 ... α N ⟩ A = N ! A ∣ α 1 α 2 ... α N ) = N ! 1 P ∑ ( − 1 ) P ∣ α P 1 α P 2 ... α P N ) = ∣ n α 1 n α 2 ... ⟩ A . Similarly, when quantum numbers α \alpha α α ′ \alpha' α ′
A ⟨ { α i } ∣ { α i ′ } ⟩ A = ∑ P ( − 1 ) P ( α 1 . . . α N ∣ α P 1 ′ . . . α P N ′ ) ⏟ d e t e r m i n a n t . _A\langle \{\alpha_i\}|\{\alpha_i'\}\rangle_A=\underbrace{\sum_{P} (-1)^{P} (\alpha_1...\alpha_N|\alpha_{P_1}'...\alpha_{P_N}')}_{determinant}. A ⟨{ α i } ∣ { α i ′ } ⟩ A = d e t er minan t P ∑ ( − 1 ) P ( α 1 ... α N ∣ α P 1 ′ ... α P N ′ ) . A ⟨ { α i } ∣ { α i ′ } ⟩ A = N ! N ! ( N ! ) − 2 ∑ P ∑ Q ( − 1 ) P + Q ( α 1 . . . ∣ P Q ∣ α 1 ′ . . . ) = ( N ! ) − 1 ∑ P ∑ P Q ( − 1 ) P Q ( α 1 . . . ∣ P Q ∣ α 1 ′ . . . ) = ∑ P ( − 1 ) P ( α 1 . . . α N ∣ α P 1 ′ . . . α P N ′ ) ⏟ d e t e r m i n a n t . \begin{aligned}
_A\langle \{\alpha_i\}|\{\alpha_i'\}\rangle_A&=\sqrt{N!}\sqrt{N!} (N!)^{-2}\sum_P\sum_Q (-1)^{P+Q}(\alpha_1...|PQ|\alpha_1'...)\\
&=(N!)^{-1}\sum_P\sum_{PQ} (-1)^{PQ} (\alpha_1...|PQ|\alpha_1'...)\\
&=\underbrace{\sum_{P} (-1)^{P} (\alpha_1...\alpha_N|\alpha_{P_1}'...\alpha_{P_N}')}_{determinant}.
\end{aligned} A ⟨{ α i } ∣ { α i ′ } ⟩ A = N ! N ! ( N ! ) − 2 P ∑ Q ∑ ( − 1 ) P + Q ( α 1 ...∣ PQ ∣ α 1 ′ ... ) = ( N ! ) − 1 P ∑ PQ ∑ ( − 1 ) PQ ( α 1 ...∣ PQ ∣ α 1 ′ ... ) = d e t er minan t P ∑ ( − 1 ) P ( α 1 ... α N ∣ α P 1 ′ ... α P N ′ ) . It is usually tidious to write ∣ { n α j } ⟩ S |\{n_{\alpha_j}\}\rangle_S ∣ { n α j } ⟩ S ∣ { n α j } ⟩ A |\{n_{\alpha_j}\}\rangle_A ∣ { n α j } ⟩ A ∣ { n α j } ⟩ |\{n_{\alpha_j}\}\rangle ∣ { n α j }⟩ N N N
Now it is time to consider the normalized wavefunction of a symmetric or antisymmetric state. The wave function is a projection to the fictitious x x x i.e.
For antisymmetric case, it is the famous Slater determinant .
ψ α 1 . . . α N ( x 1 . . . x N ) = ( x 1 . . . x N ∣ α 1 . . . α N ⟩ = 1 N ! ∑ P ( − 1 ) P ( x 1 . . . x N ∣ α P 1 ′ . . . α P N ′ ) . \psi_{\alpha_1...\alpha_N}(x_1...x_N)=(x_1...x_N|\alpha_1...\alpha_N\rangle=\frac{1}{\sqrt{N!}}\sum_{P} (-1)^{P} (x_1...x_N|\alpha_{P_1}'...\alpha_{P_N}'). ψ α 1 ... α N ( x 1 ... x N ) = ( x 1 ... x N ∣ α 1 ... α N ⟩ = N ! 1 P ∑ ( − 1 ) P ( x 1 ... x N ∣ α P 1 ′ ... α P N ′ ) . 3. Fock Space ¶ The vacuum state is
∣ 0 ⟩ ≡ ∣ { n α j = 0 } ⟩ . |0\rangle
\equiv
|\{n_{\alpha_j}=0\}\rangle. ∣0 ⟩ ≡ ∣ { n α j = 0 }⟩ . Notice that ∣ 0 ⟩ |0\rangle ∣0 ⟩
The Fock space contains sectors with different particle numbers:
F = C ⊕ H 1 ⊕ H 2 ⊕ H 3 ⊕ ⋯ . \mathcal F
=
\mathbb C
\oplus
\mathcal H_1
\oplus
\mathcal H_2
\oplus
\mathcal H_3
\oplus\cdots. F = C ⊕ H 1 ⊕ H 2 ⊕ H 3 ⊕ ⋯ . Here,
C \mathbb C C
H 1 \mathcal H_1 H 1
H 2 \mathcal H_2 H 2
and so on.
The occupation number representation of bosons/fermions are
∣ n 1 , n 2 , … ⟩ = ∏ i ( a α i † ) n α i n α i ! ∣ 0 ⟩ . |n_1,n_2,\ldots\rangle
=
\prod_i
\frac{(a_{\alpha_i}^\dagger)^{n_{\alpha_i}}}{\sqrt{n_{\alpha_i}!}}
|0\rangle. ∣ n 1 , n 2 , … ⟩ = i ∏ n α i ! ( a α i † ) n α i ∣0 ⟩ . For bosons, n α i ≥ 0 n_{\alpha_i}\ge0 n α i ≥ 0 n α i = { 0 , 1 } n_{\alpha_i}=\{0,1\} n α i = { 0 , 1 }
The total number operator will later be
N ^ = ∑ i n ^ α i . \hat N
=
\sum_i \hat n_{\alpha_i}. N ^ = i ∑ n ^ α i . The comleteness relation in H N S \mathcal H_N^S H N S
1 N S = S 1 N S = ∑ α 1 . . . α N S ∣ α 1 . . . α N ) ( α 1 . . . α N ∣ S = ∑ α 1 . . . α N n α 1 ! n α 2 ! . . . N ! ∣ α 1 . . . α N ⟩ S ⟨ α 1 . . . α N ∣ = ∑ n α 1 n α 2 . . . n α 1 ! n α 2 ! . . . N ! ∣ n a 1 n a 2 . . . ⟩ ⟨ n a 1 n a 2 . . . ∣ \begin{aligned}
\mathbf{1}_{N}^{S}&= S\mathbf 1_NS=\sum_{\alpha_1...\alpha_N} S|\alpha_1...\alpha_N)(\alpha_1...\alpha_N|S\\
&=\sum_{\alpha_1...\alpha_N}\frac{n_{\alpha_1}!n_{\alpha_2}!...}{N!}|\alpha_1...\alpha_N\rangle_S\langle \alpha_1...\alpha_N|\\
&=\sum_{ n_{\alpha_1}n_{\alpha_2}... }\frac{n_{\alpha_1}!n_{\alpha_2}!...}{N!}|n_{a_1}n_{a_2}...\rangle\langle n_{a_1}n_{a_2}...|
\end{aligned} 1 N S = S 1 N S = α 1 ... α N ∑ S ∣ α 1 ... α N ) ( α 1 ... α N ∣ S = α 1 ... α N ∑ N ! n α 1 ! n α 2 ! ... ∣ α 1 ... α N ⟩ S ⟨ α 1 ... α N ∣ = n α 1 n α 2 ... ∑ N ! n α 1 ! n α 2 ! ... ∣ n a 1 n a 2 ... ⟩ ⟨ n a 1 n a 2 ...∣ Similar relation can be constructed for fermions.
Now we should be familiar with the expression. I will use i i i α i \alpha_i α i
4. Creation and Annihilation Operators ¶ 4.1 Bosons ¶ The bosonic creation operator a i † a_i^\dagger a i † i i i
a i † ∣ n i ⟩ = n i + 1 ∣ n i + 1 ⟩ . a_i^\dagger|n_i\rangle
=
\sqrt{n_i+1}\,
|n_i+1\rangle. a i † ∣ n i ⟩ = n i + 1 ∣ n i + 1 ⟩ . The annihilation operator a i a_i a i
a i ∣ n i ⟩ = n i ∣ n i − 1 ⟩ . a_i|n_i\rangle
=
\sqrt{n_i}\,
|n_i-1\rangle. a i ∣ n i ⟩ = n i ∣ n i − 1 ⟩ . They satisfy the canonical commutation relations, [ A , B ] ≡ A B − B A [A,B]\equiv AB-BA [ A , B ] ≡ A B − B A
[ a i , a j † ] = δ i j , [a_i,a_j^\dagger]
=
\delta_{ij}, [ a i , a j † ] = δ ij , [ a i , a j ] = 0 , [a_i,a_j]
=
0, [ a i , a j ] = 0 , [ a i † , a j † ] = 0. [a_i^\dagger,a_j^\dagger]
=
0. [ a i † , a j † ] = 0. The occupation-number operator is
n ^ i = a i † a i . \hat n_i
=
a_i^\dagger a_i. n ^ i = a i † a i . Therefore,
n ^ i ∣ n i ⟩ = n i ∣ n i ⟩ . \hat n_i|n_i\rangle
=
n_i|n_i\rangle. n ^ i ∣ n i ⟩ = n i ∣ n i ⟩ . 4.2 Fermions ¶ For fermions, we introduce c i † c_i^\dagger c i † c i c_i c i
They satisfy the canonical anticommutation relations, { A , B } = A B + B A \{A,B\}=AB+BA { A , B } = A B + B A
{ c i , c j † } = δ i j , \{c_i,c_j^\dagger\}
=
\delta_{ij}, { c i , c j † } = δ ij , { c i , c j } = 0 , \{c_i,c_j\}
=
0, { c i , c j } = 0 , { c i † , c j † } = 0. \{c_i^\dagger,c_j^\dagger\}
=
0. { c i † , c j † } = 0. For i = j i=j i = j
{ c i † , c i † } = 2 ( c i † ) 2 = 0. \{c_i^\dagger,c_i^\dagger\}
=
2(c_i^\dagger)^2
=
0. { c i † , c i † } = 2 ( c i † ) 2 = 0. Therefore,
( c i † ) 2 = 0. (c_i^\dagger)^2=0. ( c i † ) 2 = 0. This immediately implies that one cannot create two identical fermions in the same single-particle state.
Hence,
The Pauli exclusion principle is therefore encoded directly in the operator algebra.
The fermionic number operator is
n ^ i = c i † c i , \hat n_i
=
c_i^\dagger c_i, n ^ i = c i † c i , and the total particle-number operator is
N ^ = ∑ i c i † c i . \hat N
=
\sum_i c_i^\dagger c_i. N ^ = i ∑ c i † c i . Quick conceptual question ¶ What is the eigenvalue of N ^ \hat N N ^
∣ 1 , 0 , 1 , 1 , 0 ⟩ ? |1,0,1,1,0\rangle? ∣1 , 0 , 1 , 1 , 0 ⟩? Since there are three occupied states,
N ^ ∣ 1 , 0 , 1 , 1 , 0 ⟩ = 3 ∣ 1 , 0 , 1 , 1 , 0 ⟩ . \hat N|1,0,1,1,0\rangle
=
3|1,0,1,1,0\rangle. N ^ ∣1 , 0 , 1 , 1 , 0 ⟩ = 3∣1 , 0 , 1 , 1 , 0 ⟩ . 5. From a Single-Particle Hamiltonian to Second Quantization ¶ This section connects directly to the tight-binding Hamiltonian introduced previously.
Suppose the single-particle Hamiltonian is
h ^ = ∑ i j h i j ∣ i ⟩ ⟨ j ∣ . \hat h
=
\sum_{ij}
h_{ij}
|i\rangle\langle j|. h ^ = ij ∑ h ij ∣ i ⟩ ⟨ j ∣. The corresponding second-quantized Hamiltonian is
H ^ = ∑ i j h i j c i † c j . \boxed{
\hat H
=
\sum_{ij}
h_{ij}
c_i^\dagger c_j.
} H ^ = ij ∑ h ij c i † c j . The operator
c i † c j c_i^\dagger c_j c i † c j has a simple physical meaning:
annihilate one particle in state j j j
create one particle in state i i i
Therefore,
∣ i ⟩ ⟨ j ∣ ⟶ c i † c j . |i\rangle\langle j|
\quad\longrightarrow\quad
c_i^\dagger c_j. ∣ i ⟩ ⟨ j ∣ ⟶ c i † c j . This is one of the most important correspondences in second quantization.
5.1 Tight-binding example ¶ Consider the one-dimensional tight-binding Hamiltonian
H = E 0 ∑ i ∣ i ⟩ ⟨ i ∣ − t ∑ i ( ∣ i ⟩ ⟨ i + 1 ∣ + ∣ i + 1 ⟩ ⟨ i ∣ ) . H
=
E_0
\sum_i
|i\rangle\langle i|
-
t
\sum_i
\left(
|i\rangle\langle i+1|
+
|i+1\rangle\langle i|
\right). H = E 0 i ∑ ∣ i ⟩ ⟨ i ∣ − t i ∑ ( ∣ i ⟩ ⟨ i + 1∣ + ∣ i + 1 ⟩ ⟨ i ∣ ) . Its second-quantized form is
H ^ = E 0 ∑ i c i † c i − t ∑ i ( c i † c i + 1 + c i + 1 † c i ) . \boxed{
\hat H
=
E_0
\sum_i
c_i^\dagger c_i
-
t
\sum_i
\left(
c_i^\dagger c_{i+1}
+
c_{i+1}^\dagger c_i
\right).
} H ^ = E 0 i ∑ c i † c i − t i ∑ ( c i † c i + 1 + c i + 1 † c i ) . The first term represents the on-site energy.
The second term represents hopping between neighboring sites.
For example,
c i + 1 † c i c_{i+1}^\dagger c_i c i + 1 † c i moves a particle from site i i i i + 1 i+1 i + 1
This illustrates an important point:
Second quantization does not replace the single-particle Hamiltonian. It lifts the single-particle operator into many-particle Fock space.
Consider the single-particle Hamiltonian in basis ∣ 1 ⟩ , ∣ 2 ⟩ , ∣ 3 ⟩ |1\rangle, |2\rangle, |3\rangle ∣1 ⟩ , ∣2 ⟩ , ∣3 ⟩
H = − t ( 0 1 0 1 0 1 0 1 0 ) . H
=
-t
\begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{pmatrix}. H = − t ⎝ ⎛ 0 1 0 1 0 1 0 1 0 ⎠ ⎞ . Write the corresponding second-quantized Hamiltonian.
The nonzero matrix elements correspond to hopping between sites 1 and 2, and between sites 2 and 3.
Therefore,
H ^ = − t ( c 1 † c 2 + c 2 † c 1 + c 2 † c 3 + c 3 † c 2 ) . \boxed{
\hat H
=
-t
\left(
c_1^\dagger c_2
+
c_2^\dagger c_1
+
c_2^\dagger c_3
+
c_3^\dagger c_2
\right).
} H ^ = − t ( c 1 † c 2 + c 2 † c 1 + c 2 † c 3 + c 3 † c 2 ) . The terms
c 2 † c 1 c_2^\dagger c_1 c 2 † c 1 and
c 2 † c 3 c_2^\dagger c_3 c 2 † c 3 can move either particle toward the middle site.
This exercise gives a direct physical interpretation of the operator products appearing in the Hamiltonian.
5.2.1 Definitions and useful commutator relations ¶ We took a rather unconventional route to introduce second quantization -- introduce the protocal first without knowning how things really work. That is, we just provide a protocal to go from single particle to many-body formulation. But why it works? We are going to address this question in this section.
To keep the notation tight, we will use the following definition of commutator
[ A , B ] ϵ = A B + ϵ B A [A,B]_{\epsilon}=AB+\epsilon BA [ A , B ] ϵ = A B + ϵ B A with fermions ϵ = + 1 \epsilon=+1 ϵ = + 1 ϵ = − 1 \epsilon=-1 ϵ = − 1
Therefore, we have
[ a α † , a β † ] ϵ = [ a α , a β ] ϵ = 0 [ a α , a β † ] ϵ = ⟨ α ∣ β ⟩ . \begin{aligned}
[a^{\dagger}_{\alpha},a^{\dagger}_{\beta}]_{\epsilon}&=[a_{\alpha},a_{\beta}]_{\epsilon}=0\\
[a_{\alpha},a^{\dagger}_{\beta}]_{\epsilon}&=\langle\alpha|\beta\rangle.
\end{aligned} [ a α † , a β † ] ϵ [ a α , a β † ] ϵ = [ a α , a β ] ϵ = 0 = ⟨ α ∣ β ⟩ . Note that we consider the case where α \alpha α β \beta β
[ a α , a β † ] ϵ = a α a β † + ϵ a β † a α → ⟨ α ∣ β ⟩ . [a_{\alpha}, a^{\dagger}_{\beta}]_{\epsilon}=a_{\alpha}a^{\dagger}_{\beta}+\epsilon a^{\dagger}_{\beta}a_{\alpha}\to \langle \alpha|\beta\rangle. [ a α , a β † ] ϵ = a α a β † + ϵ a β † a α → ⟨ α ∣ β ⟩ . Note that (64)
a α ∣ β ⟩ = a α a β † ∣ 0 ⟩ = [ a α , a β † ] ϵ ∣ 0 ⟩ − ϵ a β † a α ∣ 0 ⟩ = ⟨ α ∣ β ⟩ ∣ 0 ⟩ . \begin{aligned}
a_{\alpha}|\beta\rangle&=a_{\alpha}a^{\dagger}_{\beta}|0\rangle=[a_{\alpha},a^{\dagger}_{\beta}]_{\epsilon}|0\rangle-\epsilon a^{\dagger}_{\beta}a_{\alpha}|0\rangle\\
&=\langle \alpha|\beta\rangle|0\rangle.
\end{aligned} a α ∣ β ⟩ = a α a β † ∣0 ⟩ = [ a α , a β † ] ϵ ∣0 ⟩ − ϵ a β † a α ∣0 ⟩ = ⟨ α ∣ β ⟩ ∣0 ⟩ . Let’s summarize the chain of definition:
We define ∣ 0 ⟩ |0\rangle ∣0 ⟩ a α † a^{\dagger}_{\alpha} a α † ∣ 0 ⟩ |0\rangle ∣0 ⟩
We define a α a_{\alpha} a α ⟨ 0 \langle 0 ⟨ 0
We define the commutator (64) a α a_{\alpha} a α ∣ 0 ⟩ |0\rangle ∣0 ⟩
The notation (62) (62)
[ a α † a β , a γ † ] ϵ = a α † a β a γ † + ϵ a γ † a α † a β = a α † a β a γ † + ϵ ( − ϵ ) a α † a γ † a β = a α † [ a β , a γ † ] ϵ = − 1 . \begin{aligned}
[a^{\dagger}_{\alpha}a_{\beta},a^{\dagger}_{\gamma}]_{\epsilon}&=a^{\dagger}_{\alpha}a_{\beta}a^{\dagger}_{\gamma}+\epsilon a^{\dagger}_{\gamma}a^{\dagger}_{\alpha}a_{\beta}=a^{\dagger}_{\alpha}a_{\beta}a^{\dagger}_{\gamma}+\epsilon (-\epsilon) a^{\dagger}_{\alpha}a^{\dagger}_{\gamma}a_{\beta}\\
&=a^{\dagger}_{\alpha}[a_{\beta},a^{\dagger}_{\gamma}]_{\epsilon=-1}.
\end{aligned} [ a α † a β , a γ † ] ϵ = a α † a β a γ † + ϵ a γ † a α † a β = a α † a β a γ † + ϵ ( − ϵ ) a α † a γ † a β = a α † [ a β , a γ † ] ϵ = − 1 . This relation will be useful for later derivation.
5.2.2 The linear mapping nature between single particle states and creation/destruction operators ¶ Let the mapping from first quantization to second quantization defined as T 2 T_2 T 2
We have
T 2 ( a ∣ α ⟩ + b ∣ β ⟩ ) = a a α † ∣ 0 ⟩ + b b β † ∣ 0 ⟩ = a T 2 ( a ∣ α ⟩ ) + b T 2 ( b ∣ β ⟩ ) . T_2(a|\alpha\rangle+b|\beta\rangle)=a a^{\dagger}_{\alpha}|0\rangle+b b^{\dagger}_\beta|0\rangle=aT_2(a|\alpha\rangle)+bT_2(b|\beta\rangle). T 2 ( a ∣ α ⟩ + b ∣ β ⟩) = a a α † ∣0 ⟩ + b b β † ∣0 ⟩ = a T 2 ( a ∣ α ⟩) + b T 2 ( b ∣ β ⟩) . That suggest T 2 T_2 T 2
General expression ¶ Since the mapping from first quantization to second quantization is linear, the basis transformation in first quantization should be straight forward in the second quantization representation.
Consider a ket ∣ α ‾ ⟩ |\overline{\alpha}\rangle ∣ α ⟩
∣ α ‾ ⟩ = ∑ β ∣ β ⟩ ⟨ β ∣ U ∣ α ⟩ . |\overline{\alpha}\rangle=\sum_{\beta}|\beta\rangle\langle \beta|U|\alpha\rangle. ∣ α ⟩ = β ∑ ∣ β ⟩ ⟨ β ∣ U ∣ α ⟩ . The second quantization expression of this basis transformation is
a α ‾ † = ∑ β a β † ⟨ β ∣ U ∣ α ⟩ . a^{\dagger}_{\overline{\alpha}}=\sum_{\beta}a^{\dagger}_{\beta}\langle\beta|U|\alpha\rangle. a α † = β ∑ a β † ⟨ β ∣ U ∣ α ⟩ . Here, we express the states using the quantum number α , β \alpha,\beta α , β
Field operators ¶ However, sometimes we only know how to describe the system according to some fundamental process which is nature to express using real space information. For example, the energy might due to a specific form of the potential. In this case, it is useful to use the x x x
From basis ∣ α ⟩ |\alpha\rangle ∣ α ⟩ ∣ x ⟩ |x\rangle ∣ x ⟩
∣ α ⟩ = ∫ d x ∣ x ⟩ ⟨ x ∣ α ⟩ ≡ ∫ d x ϕ α ( x ) ∣ x ⟩ ∣ x ⟩ = ∑ α ∣ α ⟩ ⟨ α ∣ x ⟩ = ∑ α ϕ α ∗ ( x ) ∣ α ⟩ \begin{aligned}
|\alpha\rangle&=\int dx |x\rangle\langle x|\alpha\rangle\equiv\int dx \phi_{\alpha}(x)|x\rangle\\
|x\rangle&=\sum_{\alpha}|\alpha\rangle\langle\alpha|x\rangle=\sum_{\alpha}\phi_{\alpha}^*(x)|\alpha\rangle
\end{aligned} ∣ α ⟩ ∣ x ⟩ = ∫ d x ∣ x ⟩ ⟨ x ∣ α ⟩ ≡ ∫ d x ϕ α ( x ) ∣ x ⟩ = α ∑ ∣ α ⟩ ⟨ α ∣ x ⟩ = α ∑ ϕ α ∗ ( x ) ∣ α ⟩ To promote it to the field operators, we have
ψ † ( x ) = ∑ α a α † ⟨ α ∣ x ⟩ = ∑ α a α † ϕ α ∗ ( x ) ψ ( x ) = ∑ α a α ⟨ x ∣ α ⟩ = ∑ α a α ϕ α ( x ) \begin{aligned}
\psi^{\dagger}(x)&=\sum_{\alpha}a^{\dagger}_{\alpha}\langle \alpha|x\rangle=\sum_{\alpha}a^{\dagger}_{\alpha}\phi_{\alpha}^*(x)\\
\psi(x)&=\sum_{\alpha}a_{\alpha}\langle x|\alpha\rangle=\sum_{\alpha}a_{\alpha}\phi_{\alpha}(x)\\
\end{aligned} ψ † ( x ) ψ ( x ) = α ∑ a α † ⟨ α ∣ x ⟩ = α ∑ a α † ϕ α ∗ ( x ) = α ∑ a α ⟨ x ∣ α ⟩ = α ∑ a α ϕ α ( x ) 5.2.4 Symmetry actions ¶ Why we care about the basis transformation? Because it is related to how symmetries act on states. In general, the symmetry operator acts on states projectively , U g U h = ω ( g , h ) U g h U_gU_h=\omega(g,h)U_{gh} U g U h = ω ( g , h ) U g h ω ( g , h ) \omega(g,h) ω ( g , h ) linearly . However, here, we just keep our mind simple by assuming the vacuum to be unique . Notice that this is the place where the distinction of vacuum of some second quantized operator and the many-body ground state becomes important. There are cases that the many-body ground state manifold has dimension higher than 1. We will skip related discussion for now and focus on the simple case that the ground state is unique and is the vacuum for the corresponding destruction operator.
Since the vacuum is unique, we found the action of symmetry can always be linear after the redefinition of the phase factor.
Suppose we have
U g ∣ 0 ⟩ = e i ϕ ( g ) ∣ 0 ⟩ . U_g|0\rangle=e^{i\phi(g)}|0\rangle. U g ∣0 ⟩ = e i ϕ ( g ) ∣0 ⟩ . Then
U g U h ∣ 0 ⟩ = e i [ ϕ ( g ) + ϕ ( h ) ] ∣ 0 ⟩ . U_gU_h|0\rangle=e^{i[\phi(g)+\phi(h)]}|0\rangle. U g U h ∣0 ⟩ = e i [ ϕ ( g ) + ϕ ( h )] ∣0 ⟩ . By definition, we also have
U g h ∣ 0 ⟩ = e i ϕ ( g h ) ∣ 0 ⟩ . U_{gh}|0\rangle=e^{i\phi(gh)}|0\rangle. U g h ∣0 ⟩ = e i ϕ ( g h ) ∣0 ⟩ . By redefine the operator U g ′ = e − i ϕ ( g ) U g U'_g=e^{-i\phi(g)}U_g U g ′ = e − i ϕ ( g ) U g U g ′ ∣ 0 ⟩ = ∣ 0 ⟩ U_g'|0\rangle=|0\rangle U g ′ ∣0 ⟩ = ∣0 ⟩ ω ( g , h ) = 1 \omega(g,h)=1 ω ( g , h ) = 1
What goes wrong when we have degeneracy in the ground state is that the symmetry action on one of the degenerated state may couple other degenerated states and the starting point (72)
Consider a one-particle symmetry acts as
∣ ψ ⟩ ↦ U ∣ ψ ⟩ . |\psi\rangle\mapsto U|\psi\rangle. ∣ ψ ⟩ ↦ U ∣ ψ ⟩ . Charge conjugation ¶ Parity ¶ Time-reversal ¶ 5.3 General construction of operators from first quantization to second quantization ¶ 5.3.1 Representation of symmetric operators in Fock space. ¶ 5.3.2 Single-particle operators in second quantization ¶ 5.3.3 Two-particle operators in second quantization ¶ 6. Why Second Quantization Becomes Essential for Interactions ¶ For noninteracting particles, second quantization is elegant.
For interacting particles, it becomes especially powerful.
A generic first-quantized Hamiltonian can be written schematically as
H = ∑ i h ( r i ) + 1 2 ∑ i ≠ j V ( r i − r j ) . H
=
\sum_i
h(\mathbf r_i)
+
\frac{1}{2}
\sum_{i\neq j}
V(\mathbf r_i-\mathbf r_j). H = i ∑ h ( r i ) + 2 1 i = j ∑ V ( r i − r j ) . The second-quantized form is
H = ∑ i j h i j c i † c j + 1 2 ∑ i j k l V i j k l c i † c j † c l c k . \boxed{
H
=
\sum_{ij}
h_{ij}
c_i^\dagger c_j
+
\frac{1}{2}
\sum_{ijkl}
V_{ijkl}
c_i^\dagger
c_j^\dagger
c_l
c_k.
} H = ij ∑ h ij c i † c j + 2 1 ijk l ∑ V ijk l c i † c j † c l c k . The one-body term
c i † c j c_i^\dagger c_j c i † c j describes the motion of one particle.
The two-body term
c i † c j † c l c k c_i^\dagger
c_j^\dagger
c_l
c_k c i † c j † c l c k describes a scattering process:
( k , l ) → ( i , j ) . (k,l)
\rightarrow
(i,j). ( k , l ) → ( i , j ) . The operators
remove two particles from the initial states, while
c i † c j † c_i^\dagger c_j^\dagger c i † c j † create two particles in the final states.
Thus interactions can be interpreted directly as microscopic scattering processes.
8. Example: The Hubbard Model ¶ One of the simplest interacting lattice Hamiltonians is the Hubbard model:
H = − t ∑ ⟨ i j ⟩ , σ ( c i σ † c j σ + c j σ † c i σ ) + U ∑ i n i ↑ n i ↓ . \boxed{
H
=
-t
\sum_{\langle ij\rangle,\sigma}
\left(
c_{i\sigma}^\dagger c_{j\sigma}
+
c_{j\sigma}^\dagger c_{i\sigma}
\right)
+
U
\sum_i
n_{i\uparrow}n_{i\downarrow}.
} H = − t ⟨ ij ⟩ , σ ∑ ( c iσ † c jσ + c jσ † c iσ ) + U i ∑ n i ↑ n i ↓ . The hopping term,
− t ∑ ⟨ i j ⟩ , σ c i σ † c j σ , -t
\sum_{\langle ij\rangle,\sigma}
c_{i\sigma}^\dagger c_{j\sigma}, − t ⟨ ij ⟩ , σ ∑ c iσ † c jσ , favors particle delocalization.
The interaction term,
U ∑ i n i ↑ n i ↓ , U
\sum_i
n_{i\uparrow}n_{i\downarrow}, U i ∑ n i ↑ n i ↓ , assigns an energy cost to double occupation.
The physics therefore involves competition between
kinetic delocalization and local interaction . \boxed{
\text{kinetic delocalization}
\quad\text{and}\quad
\text{local interaction}.
} kinetic delocalization and local interaction . A large fraction of condensed matter physics can be viewed as understanding the consequences of such competing terms.
9. From Microscopic Particles to Collective Excitations ¶ Once many particles interact, the microscopic particles are not always the most useful variables for describing low-energy physics.
Examples include
atoms → phonons , \text{atoms}
\rightarrow
\text{phonons}, atoms → phonons , spins → magnons , \text{spins}
\rightarrow
\text{magnons}, spins → magnons , electrons → density fluctuations , \text{electrons}
\rightarrow
\text{density fluctuations}, electrons → density fluctuations , interacting electrons → quasiparticles , \text{interacting electrons}
\rightarrow
\text{quasiparticles}, interacting electrons → quasiparticles , paired electrons → Bogoliubov quasiparticles . \text{paired electrons}
\rightarrow
\text{Bogoliubov quasiparticles}. paired electrons → Bogoliubov quasiparticles . The microscopic Hamiltonian may be written using operators such as
c i , c i † , c_i,
\qquad
c_i^\dagger, c i , c i † , but the effective low-energy Hamiltonian may take the form
H e f f = ∑ q ω q b q † b q . H_{\mathrm{eff}}
=
\sum_q
\omega_q
b_q^\dagger b_q. H eff = q ∑ ω q b q † b q . The operator b q † b_q^\dagger b q †
This is one of the central ideas of condensed matter physics:
The elementary excitations of an interacting many-body system need not be the microscopic particles from which the system is built.
10. Minimal Example: Phonons ¶ Consider a one-dimensional chain of atoms with displacement u j u_j u j p j p_j p j
The harmonic-chain Hamiltonian is
H = ∑ j p j 2 2 m + K 2 ∑ j ( u j + 1 − u j ) 2 . H
=
\sum_j
\frac{p_j^2}{2m}
+
\frac{K}{2}
\sum_j
(u_{j+1}-u_j)^2. H = j ∑ 2 m p j 2 + 2 K j ∑ ( u j + 1 − u j ) 2 . Here,
Introduce normal-mode coordinates:
u j = 1 N ∑ q u q e i q R j , u_j
=
\frac{1}{\sqrt N}
\sum_q
u_q
e^{iqR_j}, u j = N 1 q ∑ u q e i q R j , and
p j = 1 N ∑ q p q e i q R j . p_j
=
\frac{1}{\sqrt N}
\sum_q
p_q
e^{iqR_j}. p j = N 1 q ∑ p q e i q R j . The Hamiltonian becomes a sum over independent momentum modes:
H = ∑ q [ p q p − q 2 m + m ω q 2 2 u q u − q ] . H
=
\sum_q
\left[
\frac{p_qp_{-q}}{2m}
+
\frac{m\omega_q^2}{2}
u_qu_{-q}
\right]. H = q ∑ [ 2 m p q p − q + 2 m ω q 2 u q u − q ] . The dispersion relation is
ω q = 2 K m ∣ sin ( q a 2 ) ∣ . \boxed{
\omega_q
=
2\sqrt{\frac{K}{m}}
\left|
\sin\left(\frac{qa}{2}\right)
\right|.
} ω q = 2 m K ∣ ∣ sin ( 2 q a ) ∣ ∣ . Each momentum mode behaves like an independent harmonic oscillator.
10.2 Quantization of the normal modes ¶ For each mode, define bosonic creation and annihilation operators.
Schematically,
u q ∝ b q + b − q † , u_q
\propto
b_q+b_{-q}^\dagger, u q ∝ b q + b − q † , and
p q ∝ b q − b − q † . p_q
\propto
b_q-b_{-q}^\dagger. p q ∝ b q − b − q † . The Hamiltonian becomes
H = ∑ q ℏ ω q ( b q † b q + 1 2 ) . \boxed{
H
=
\sum_q
\hbar\omega_q
\left(
b_q^\dagger b_q
+
\frac{1}{2}
\right).
} H = q ∑ ℏ ω q ( b q † b q + 2 1 ) . The occupation number
n q = b q † b q n_q
=
b_q^\dagger b_q n q = b q † b q counts the number of phonons in mode q q q
The original microscopic degrees of freedom were atomic displacements.
The natural quantum excitations are now phonons.
Thus,
coupled atomic motion → independent collective modes → phonons . \boxed{
\text{coupled atomic motion}
\rightarrow
\text{independent collective modes}
\rightarrow
\text{phonons}.
} coupled atomic motion → independent collective modes → phonons . This provides a concrete example of emergence.
11. Key Conceptual Message ¶ The phonon example illustrates why second quantization and collective excitations naturally belong together.
The microscopic system may contain a very large number of interacting degrees of freedom.
However, after identifying the appropriate normal modes, the Hamiltonian may take the simple form
H = ∑ q ϵ q γ q † γ q . H
=
\sum_q
\epsilon_q
\gamma_q^\dagger\gamma_q. H = q ∑ ϵ q γ q † γ q . The operators γ q † \gamma_q^\dagger γ q † appropriate emergent excitations rather than necessarily the original microscopic particles.
A useful perspective for the rest of condensed matter physics is:
Much of condensed matter physics is the search for the right creation and annihilation operators.
12. Learning Outcomes ¶ By the end of Week 2, students should be able to:
Translate between particle-coordinate and occupation-number descriptions.
Explain the meaning of Fock space.
Use bosonic commutation relations,
[ a i , a j † ] = δ i j , [a_i,a_j^\dagger]=\delta_{ij}, [ a i , a j † ] = δ ij , and fermionic anticommutation relations,
{ c i , c j † } = δ i j . \{c_i,c_j^\dagger\}=\delta_{ij}. { c i , c j † } = δ ij . Explain how the Pauli exclusion principle follows from
( c i † ) 2 = 0. (c_i^\dagger)^2=0. ( c i † ) 2 = 0. Convert a single-particle matrix Hamiltonian
into the second-quantized form
∑ i j h i j c i † c j . \sum_{ij}
h_{ij}
c_i^\dagger c_j. ij ∑ h ij c i † c j . Interpret
c i † c j c_i^\dagger c_j c i † c j as a one-particle transition.
Interpret
c i † c j † c l c k c_i^\dagger c_j^\dagger c_l c_k c i † c j † c l c k as a two-particle scattering process.
Explain why collective excitations can be treated as particles even though they arise from coordinated motion of many microscopic degrees of freedom.
13. Final Summary ¶ The progression of the lecture can be summarized as
First quantization Ψ ( x 1 , … , x N ) \boxed{
\text{First quantization}
\quad
\Psi(x_1,\ldots,x_N)
} First quantization Ψ ( x 1 , … , x N ) Occupation numbers ∣ n 1 , n 2 , … ⟩ \boxed{
\text{Occupation numbers}
\quad
|n_1,n_2,\ldots\rangle
} Occupation numbers ∣ n 1 , n 2 , … ⟩ a i † , a i \boxed{
a_i^\dagger,\ a_i
} a i † , a i H = ∑ i j t i j a i † a j + ∑ i j k l V i j k l a i † a j † a l a k \boxed{
H
=
\sum_{ij}
t_{ij}a_i^\dagger a_j
+
\sum_{ijkl}
V_{ijkl}
a_i^\dagger a_j^\dagger a_l a_k
} H = ij ∑ t ij a i † a j + ijk l ∑ V ijk l a i † a j † a l a k Emergent excitations H e f f = ∑ q ϵ q γ q † γ q + ⋯ \boxed{
\text{Emergent excitations}
\quad
H_{\mathrm{eff}}
=
\sum_q
\epsilon_q
\gamma_q^\dagger\gamma_q
+\cdots
} Emergent excitations H eff = q ∑ ϵ q γ q † γ q + ⋯ The essential conceptual message is:
Second quantization provides the natural language for describing occupation, interactions, and emergent collective excitations in quantum many-body systems.
14. Connection to the Next Lecture ¶ In the next lecture, Tight-Binding Models and Bloch Theorem , we will use the second-quantized language developed here to study translationally invariant lattice systems.
A central transformation will be
c k = 1 N ∑ j e − i k R j c j , c_k
=
\frac{1}{\sqrt N}
\sum_j
e^{-ikR_j}
c_j, c k = N 1 j ∑ e − ik R j c j , which reorganizes the lattice degrees of freedom into momentum-space modes.
This leads naturally to
The sequence of the first three weeks is therefore
Week 1: Why many-body physics? \boxed{
\text{Week 1: Why many-body physics?}
} Week 1: Why many-body physics? Week 2: What language should we use? \boxed{
\text{Week 2: What language should we use?}
} Week 2: What language should we use? Week 3: How does symmetry organize particle motion? \boxed{
\text{Week 3: How does symmetry organize particle motion?}
} Week 3: How does symmetry organize particle motion?
Blaizot, J.-P., & Ripka, G. (1986). Quantum theory of finite systems. (No Title) .