Quantum Mechanics

Second Quantization

Second Quantization

Quantum Many-Body States

Identical Particles

Classical mechanics: each particle is labeled by a distinct position (Subscript[r, i]) => any different configuration of {(Subscript[r, i])} correspond to a different classical many-body state.

Quantum mechanics: particles are identical, such that exchanging two particles ((Subscript[r, i])↔(Subscript[r, j])) does not lead to a different quantum many-body state.

p(…,(Subscript[r, i]),…,(Subscript[r, j]),…)=p(…,(Subscript[r, j]),…,(Subscript[r, i]),…),

p(…,(Subscript[r, i]),…,(Subscript[r, j]),…)=((|Ψ(…,(Subscript[r, i]),…,(Subscript[r, j]),…)|)^2).

Ψ(…,(Subscript[r, i]),…,(Subscript[r, j]),…)=(e^(i φ))Ψ(…,(Subscript[r, j]),…,(Subscript[r, i]),…).

trivial representation => bosons

(Subscript[Ψ, B])(…,(Subscript[r, i]),…,(Subscript[r, j]),…)=+(Subscript[Ψ, B])(…,(Subscript[r, j]),…,(Subscript[r, i]),…)

sign representation => fermions

(Subscript[Ψ, F])(…,(Subscript[r, i]),…,(Subscript[r, j]),…)=-(Subscript[Ψ, F])(…,(Subscript[r, j]),…,(Subscript[r, i]),…)

Dirac Notations

(Ket[α])=∫(d^d)r (Subscript[ψ, α])(r)(Ket[r]),

A two-particle state with the 1st particle in (Ket[(Subscript[α, 1])]) and the 2nd particle in (Ket[(Subscript[α, 2])]) will be described by

(Ket[(Subscript[α, 1])])⊗(Ket[(Subscript[α, 2])])=∫(d^d)(Subscript[r, 1])∫(d^d)(Subscript[r, 2]) (Subscript[ψ, (Subscript[α, 1])])((Subscript[r, 1]))(Subscript[ψ, (Subscript[α, 2])])((Subscript[r, 2]))(Ket[(Subscript[r, 1])])⊗(Ket[(Subscript[r, 2])])
=∫(d^d)(Subscript[r, 1])∫(d^d)(Subscript[r, 2]) Ψ((Subscript[r, 1]),(Subscript[r, 2]))(Ket[(Subscript[r, 1])])⊗(Ket[(Subscript[r, 2])]).

Ψ((Subscript[r, 1]),(Subscript[r, 2]))=(Subscript[ψ, (Subscript[α, 1])])((Subscript[r, 1]))(Subscript[ψ, (Subscript[α, 2])])((Subscript[r, 2])) is identified as the two-body wave function.

Exchanging (Subscript[r, 1])↔(Subscript[r, 2]) in the wave function Ψ((Subscript[r, 1]),(Subscript[r, 2])) leads to a new wave function Ψ'((Subscript[r, 1]),(Subscript[r, 2]))

Ψ'((Subscript[r, 1]),(Subscript[r, 2]))=Ψ((Subscript[r, 2]),(Subscript[r, 1]))=(Subscript[ψ, (Subscript[α, 1])])((Subscript[r, 2]))(Subscript[ψ, (Subscript[α, 2])])((Subscript[r, 1]))=(Subscript[ψ, (Subscript[α, 2])])((Subscript[r, 1]))(Subscript[ψ, (Subscript[α, 1])])((Subscript[r, 2])),

which corresponds to a new state

∫(d^d)(Subscript[r, 1])∫(d^d)(Subscript[r, 2]) Ψ'((Subscript[r, 1]),(Subscript[r, 2]))(Ket[(Subscript[r, 1])])⊗(Ket[(Subscript[r, 2])])
=∫(d^d)(Subscript[r, 1])∫(d^d)(Subscript[r, 2])(Subscript[ψ, (Subscript[α, 2])])((Subscript[r, 1]))(Subscript[ψ, (Subscript[α, 1])])((Subscript[r, 2]))(Ket[(Subscript[r, 1])])⊗(Ket[(Subscript[r, 2])])
=(Ket[(Subscript[α, 2])])⊗(Ket[(Subscript[α, 1])]),

describing a two-particle state with the 1st particle in (Ket[(Subscript[α, 2])]) and the 2nd particle in (Ket[(Subscript[α, 1])]).

First-Quantized States

Suppose the single-particle Hilbert space is D dimensional, spanned by the single-particle basis states (Ket[α]) (α=1,2,…,D).

The many-body Hilbert space of N particles will be (D^N) dimensional, spanned by the many-body basis states

where (Subscript[α, i])=1,2,…,D labels the state of the ith particle.

A generic first-quantized state is a linear superposition of these basis states

(Ket[Ψ])=(∑_([α]))Ψ[α](Ket[[α]]),

where the coefficient Ψ[α]∈ℂ is also called the many-body wave function (as a more general function of labels (Subscript[α, i]) not positions (Subscript[r, i])).

For identical bosons, Ψ[α] must be symmetric

(Subscript[Ψ, B])(…,(Subscript[α, i]),…,(Subscript[α, j]),…)=+(Subscript[Ψ, B])(…,(Subscript[α, j]),…,(Subscript[α, i]),…)

For identical fermions, Ψ[α] must be antisymmetric

(Subscript[Ψ, F])(…,(Subscript[α, i]),…,(Subscript[α, j]),…)=-(Subscript[Ψ, F])(…,(Subscript[α, j]),…,(Subscript[α, i]),…)

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bosonic states by symmetrization

fermionic states by antisymmetrization

(((-))^π)=({{{, ({{+1, if π has even number of inversions}, {-1, if π has odd number of inversions}})}})

({{π(123), 123, 231, 312, 321, 213, 132}, {(((-))^π), +1, +1, +1, -1, -1, -1}})

Examples of bosonic states (unnormalized):

𝒮(Ket[α])⊗(Ket[β])=(Ket[α])⊗(Ket[β])+(Ket[β])⊗(Ket[α]), (assuming α≠β)
𝒮(Ket[α])⊗(Ket[α])=(Ket[α])⊗(Ket[α]).

Examples of fermionic states (unnormalized):

𝒜(Ket[α])⊗(Ket[β])=(Ket[α])⊗(Ket[β])-(Ket[β])⊗(Ket[α]), (assuming α≠β)
𝒜(Ket[α])⊗(Ket[α])=0 => no such fermionic state.

Pauli exclusion principle: two (or more) identical fermions can not occupy the same state simultaneously.

𝒮(Ket[α])⊗(Ket[β])=𝒮(Ket[β])⊗(Ket[α]),
𝒜(Ket[α])⊗(Ket[β])=-𝒜(Ket[β])⊗(Ket[α]).

The first-quantized Hilbert space is redundant => there are fewer basis states in the bosonic and fermionic subspaces.

Consider N particles, each can take one of D different single-particle states,

the dimension of bosonic subspace:

(Subscript[𝒟, B])=(((N+D-1)!)/(N!(D-1)!)).

the dimension of fermionic subspace:

(Subscript[𝒟, F])=((D!)/(N!(D-N)!)).

It turns out that (Subscript[𝒟, B])+(Subscript[𝒟, F])≤(D^N) as long as N>1 => the remaining basis states in the first-quantized Hilbert space are unphysical (for identical particles).

Second-Quantized States (Fock States)

In first-quantization, we ask: Which particle is in which state?

In second-quantization, we ask: How many particles are there in every state?

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Each basis state in the many-body Hilbert space is labeled by a set of occupation numbers (Subscript[n, α]) (for α=1,2,…,D)

meaning that there are (Subscript[n, α]) particles in the state (Ket[α]).

(Subscript[n, α])=({{{, ({{0,1,2,3,…, bosons,}, {0,1, fermions.}})}})

For bosons, (Subscript[n, α]) can be any non-negative integer.

For fermions, (Subscript[n, α]) can only take 0 or 1, due to the Pauli exclusion principle.

The occupation numbers (Subscript[n, α]) sum up to the total number of particles, i.e. (∑_α)(Subscript[n, α])=N.

The states (Ket[[n]]) are also known as Fock states.

All Fock states form a complete set of basis for the many-body Hilbert space, or the Fock space.

Any generic second-quantized many-body state is a linear combination of Fock states,

(Ket[Ψ])=(∑_([n]))Ψ[n](Ket[[n]]).

Representation of Fock States

The Fock state with all occupation numbers to be zero is called the vacuum state, denoted as

(Ket[0])≡(Ket[…,0,…])

It corresponds to the tensor product unit in the first-quantization, which can be written as

(Subscript[(Ket[0]), B])=(Subscript[(Ket[0]), F])=\[ScriptOne].

We use a subscript B/F to indicate whether the Fock state is bosonic (B) or fermionic (F). For vacuum state, there is no difference between them.

The Fock state with only one non-zero occupation number is a single-mode Fock state, denoted as

(Ket[(Subscript[n, α])])=(Ket[…,0,(Subscript[n, α]),0,…])

In terms of the first-quantized states

(Subscript[(Ket[(Subscript[1, α])]), B])=(Subscript[(Ket[(Subscript[1, α])]), F])=(Ket[α]),
(Subscript[(Ket[(Subscript[2, α])]), B])=(Ket[α])⊗(Ket[α]),
(Subscript[(Ket[(Subscript[3, α])]), B])=(Ket[α])⊗(Ket[α])⊗(Ket[α]),
(Subscript[(Ket[(Subscript[n, α])]), B])=(Underscript[(Underscript[(Ket[α])⊗(Ket[α])⊗…⊗(Ket[α]), ⏟]), (Subscript[n, α]) factors])≡((Ket[α])^(⊗(Subscript[n, α]))).

For multi-mode Fock states (meaning more than one single-particle state (Ket[α]) is involved), the first-quantized state will involve appropriate symmetrization depending on the particle statistics. For example,

(Subscript[(Ket[(Subscript[1, α]),(Subscript[1, β])]), B])=(1/(Sqrt[2]))((Ket[α])⊗(Ket[β])+(Ket[β])⊗(Ket[α])),
(Subscript[(Ket[(Subscript[1, α]),(Subscript[1, β])]), F])=(1/(Sqrt[2]))((Ket[α])⊗(Ket[β])-(Ket[β])⊗(Ket[α])).

Note the difference between bosonic and fermionic Fock states (even if their occupation numbers are the same). Here are more examples

(Subscript[(Ket[(Subscript[2, α]),(Subscript[1, β])]), B])=(1/(Sqrt[3]))((Ket[α])⊗(Ket[α])⊗(Ket[β])+(Ket[α])⊗(Ket[β])⊗(Ket[α])+(Ket[β])⊗(Ket[α])⊗(Ket[α])),
(Subscript[(Ket[(Subscript[1, α]),(Subscript[1, β]),(Subscript[1, γ])]), F])=(1/(Sqrt[6]))((Ket[α])⊗(Ket[β])⊗(Ket[γ])+(Ket[β])⊗(Ket[γ])⊗(Ket[α])+(Ket[γ])⊗(Ket[α])⊗(Ket[β])-(Ket[γ])⊗(Ket[β])⊗(Ket[α])-(Ket[β])⊗(Ket[α])⊗(Ket[γ])-(Ket[α])⊗(Ket[γ])⊗(Ket[β])).

for bosons,

for fermions,

Creation and Annihilation Operators

State Insertion and Deletion

Let (Ket[α]), (Ket[β]) be single-particle states.

Let \[ScriptOne] be the tensor identity (meaning that (Ket[α])⊗\[ScriptOne]=\[ScriptOne]⊗(Ket[α])=(Ket[α])).

Let (Ket[Ψ]), (Ket[Φ]) be generic first-quantized states as in Eq. ().

Linearity (for a,b∈ℂ)

(Ket[α])(Subscript[⊳, ±])(a(Ket[Ψ])+b(Ket[Φ]))=a(Ket[α])(Subscript[⊳, ±])(Ket[Ψ])+b(Ket[α])(Subscript[⊳, ±])(Ket[Φ]),
(Ket[α])(Subscript[⊲, ±])(a(Ket[Ψ])+b(Ket[Φ]))=a(Ket[α])(Subscript[⊲, ±])(Ket[Ψ])+b(Ket[α])(Subscript[⊲, ±])(Ket[Φ]).

Vacuum action

(Ket[α])(Subscript[⊳, ±])\[ScriptOne]=(Ket[α]),
(Ket[α])(Subscript[⊲, ±])\[ScriptOne]=0.

Recursive relation

(Ket[α])(Subscript[⊳, ±])(Ket[β])⊗(Ket[Ψ])=(Ket[α])⊗(Ket[β])⊗(Ket[Ψ])±(Ket[β])⊗((Ket[α])(Subscript[⊳, ±])(Ket[Ψ])),
(Ket[α])(Subscript[⊲, ±])(Ket[β])⊗(Ket[Ψ])=(BraKet[α, β])(Ket[Ψ])±(Ket[β])⊗((Ket[α])(Subscript[⊲, ±])(Ket[Ψ])).

Boson Creation and Annihilation

The boson creation operator (Subscript[b, α]^†) adds a boson to the single-particle state (Ket[α]), increasing the occupation number by one (Subscript[n, α])→(Subscript[n, α])+1. It acts on a N-particle first-quantized state (Ket[Ψ]) as

where (Ket[α])(SubPlus[⊳]) inserts the single-particle state (Ket[α]) to N+1 possible insertion positions symmetrically.

The boson annihilation operator (Subscript[b, α]) removes a boson from the single-particle state (Ket[α]), reducing the occupation number by one (Subscript[n, α])->(Subscript[n, α])-1 (while (Subscript[n, α])>0). It acts on a N-particle first-quantized state (Ket[Ψ]) as

where (Ket[α])(SubPlus[⊲]) removes the single-particle state (Ket[α]) from N possible deletion positions symmetrically.

Single-Mode Fock States

(Subscript[b, α]^†)(Ket[(Subscript[n, α])])=(1/(Sqrt[(Subscript[n, α])+1]))(Ket[α])(SubPlus[⊳])((Ket[α])^(⊗(Subscript[n, α])))
=(((Subscript[n, α])+1)/(Sqrt[(Subscript[n, α])+1]))((Ket[α])^(⊗((Subscript[n, α])+1)))
=(Sqrt[(Subscript[n, α])+1])(Ket[(Subscript[n, α])+1]).

(Subscript[b, α])(Ket[(Subscript[n, α])])=(1/(Sqrt[(Subscript[n, α])]))(Ket[α])(SubPlus[⊲])((Ket[α])^(⊗(Subscript[n, α])))
=((Subscript[n, α])/(Sqrt[(Subscript[n, α])]))((Ket[α])^(⊗((Subscript[n, α])-1)))
=(Sqrt[(Subscript[n, α])])(Ket[(Subscript[n, α])-1]).

Especially, when acting on the vacuum state

(Subscript[b, α]^†)(Ket[(Subscript[0, α])])=(Ket[(Subscript[1, α])]),
(Subscript[b, α])(Ket[(Subscript[0, α])])=0.

Using Eq. (), we can show that

meaning that (Subscript[b, α]^†)(Subscript[b, α]) is the boson number operator of the (Ket[α]) state.

Generic Fock States

Operator Identities

Fermion Creation and Annihilation

The fermion creation operator (Subscript[c, α]^†) adds a fermion to the single-particle state (Ket[α]), increasing the occupation number by one (Subscript[n, α])→(Subscript[n, α])+1 (while (Subscript[n, α])=0). It acts on a N-particle first-quantized state (Ket[Ψ]) as

where (Ket[α])(SubMinus[⊳]) inserts the single-particle state (Ket[α]) to N+1 possible insertion positions anti-symmetrically.

The fermion annihilation operator (Subscript[c, α]) removes a fermion from the single-particle state (Ket[α]), reducing the occupation number by one (Subscript[n, α])->(Subscript[n, α])-1 (while (Subscript[n, α])=1). It acts on a N-particle first-quantized state (Ket[Ψ]) as

where (Ket[α])(SubMinus[⊲]) removes the single-particle state (Ket[α]) from N possible deletion positions anti-symmetrically.

Single-Mode Fock States

(Subscript[c, α]^†)(Ket[(Subscript[0, α])])=(Ket[α])(SubMinus[⊳])\[ScriptOne]=(Ket[α])=(Ket[(Subscript[1, α])])
(Subscript[c, α]^†)(Ket[(Subscript[1, α])])=(1/(Sqrt[2]))(Ket[α])(SubMinus[⊳])(Ket[α])=(1/(Sqrt[2]))((Ket[α])⊗(Ket[α])-(Ket[α])⊗(Ket[α]))=0

(Subscript[c, α])(Ket[(Subscript[0, α])])=0
(Subscript[c, α])(Ket[(Subscript[1, α])])=(Ket[α])(SubMinus[⊲])(Ket[α])=\[ScriptOne]=(Ket[(Subscript[0, α])]).

Using Eq. (), we can show that

meaning that (Subscript[c, α]^†)(Subscript[c, α]) is the fermion number operator of the (Ket[α]) state.

Generic Fock States

Operator Identities