130A Quantum Physics

Part 3. Quantum Statistics

Introduction

Tensors are Vectors

From One to Two

({{ , , (Subscript[qubit, B]), }, {, , (Ket[0]), (Ket[1])}, {(Subscript[qubit, A]), (Ket[0]), (Ket[00]), (Ket[01])}, {, (Ket[1]), (Ket[10]), (Ket[11])}})

A generic two-qubit quantum state will be a linear superposition of these basis states

(Ket[ψ])=(Subscript[ψ, 00])(Ket[00])+(Subscript[ψ, 01])(Ket[01])+(Subscript[ψ, 10])(Ket[10])+(Subscript[ψ, 11])(Ket[11]),

where the coefficients (Subscript[ψ, α β]) is most naturally arranged as a 2×2 array (a matrix) like

(({{(Subscript[ψ, 00]), (Subscript[ψ, 01])}, {(Subscript[ψ, 10]), (Subscript[ψ, 11])}})).

However, it makes no difference to rearrange them in a vector

(({{(Subscript[ψ, 00])}, {(Subscript[ψ, 01])}, {(Subscript[ψ, 10])}, {(Subscript[ψ, 11])}}))→(({{(Subscript[ψ, 0])}, {(Subscript[ψ, 1])}, {(Subscript[ψ, 2])}, {(Subscript[ψ, 3])}})).

We can relabel the index (Subscript[ψ, α β])→(Subscript[ψ, i]) by converting each binary string α β to an integer i (e.g. through the binary number encoding).

A matrix can be viewed as a vector by flattening. Here, (ℂ^(2×2))→(ℂ^4).

In vector representation, the ket vector (Ket[00]) is a tensor product of (Subscript[(Ket[0]), A]) and (Subscript[(Ket[0]), B]),

(Ket[00])=(Subscript[(Ket[0]), A])⊗(Subscript[(Ket[0]), B])≏(({{1}, {0}}))⊗(({{1}, {0}}))=(({{1}, {0}, {0}, {0}})).

Similarly,

(Ket[01])=(Subscript[(Ket[0]), A])⊗(Subscript[(Ket[1]), B])≏(({{1}, {0}}))⊗(({{0}, {1}}))=(({{0}, {1}, {0}, {0}})),
(Ket[10])=(Subscript[(Ket[1]), A])⊗(Subscript[(Ket[0]), B])≏(({{0}, {1}}))⊗(({{1}, {0}}))=(({{0}, {0}, {1}, {0}})),
(Ket[11])=(Subscript[(Ket[1]), A])⊗(Subscript[(Ket[1]), B])≏(({{0}, {1}}))⊗(({{0}, {1}}))=(({{0}, {0}, {0}, {1}})).

From Two to Many

Each basis state (Ket[α]) is labeled by a bit string α∈(({0,1})^(×N)),

α=(Subscript[α, 1])(Subscript[α, 2])(Subscript[…α, N])   for (Subscript[α, i])∈{0,1},

and defined by the tensor product of single-qubit states

(Ket[α]):=(Ket[(Subscript[α, 1])])⊗(Ket[(Subscript[α, 2])])⊗…⊗(Ket[(Subscript[α, N])]).

A generic N-qubit state will be a linear combination of all multi-qubit basis states

(Ket[Ψ])=(∑_α)(Subscript[Ψ, α])(Ket[α]).

The coefficients (Subscript[Ψ, α]) form a (ℂ^(2×2×…×2)) tensor, but can also be viewed as a (ℂ^(2^N))vector by flattening.

Quantum Many-Body States

Overview

Distinct particles: spins, qubits ...

Identical particles: bosons, fermions ...

Distinct Particles

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

The many-body Hilbert space of N distinct 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 many-body state is a linear superposition of these basis states

The coefficient (Subscript[Ψ, α]) is also called the many-body wave function.

The probability to find the many-body system in a specific state (Ket[α]) is given by

Identical Particles

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p((Subscript[α, 1])(Subscript[α, 2])|Ψ)=p((Subscript[α, 2])(Subscript[α, 1])|Ψ)

Each permutation π∈(Subscript[S, N]) is a bijective (invertible) map from N objects to themselves. For example,

123(Overscript[→, π])132

is a permutation in (Subscript[S, 3]), defined by π(1)=1, π(2)=3, π(3)=2.

(Subscript[α, π]) denotes a new sequence obtained from the sequence α by permuting its elements by π. For example,

α=(Subscript[α, 1])(Subscript[α, 2])(Subscript[α, 3])(Overscript[→, π])(Subscript[α, π])=(Subscript[α, 1])(Subscript[α, 3])(Subscript[α, 2]).

(Subscript[(OverHat[𝒫]), π]) denotes the operator that take the state (Ket[α]) to (Ket[(Subscript[α, π])]) for all α, which implements the permutation of particles.

∀π∈(Subscript[S, N]):p(α|Ψ)=p((Subscript[α, π])|Ψ)

∀π∈(Subscript[S, N]):((|(Subscript[Ψ, α])|)^2)=((|(Subscript[Ψ, (Subscript[α, π])])|)^2).

(Subscript[Ψ, α])=(e^(i φ))(Subscript[Ψ, (Subscript[α, π])]).

symmetric (trivial) representation => bosons

antisymmetric (sign) representation => fermions

where (((-))^π) denotes the permutation sign of π

(((-))^π)=({{{, ({{+1, if π contains even number of exchanges,}, {-1, if π contains odd number of exchanges.}})}})

Take the (Subscript[S, 3]) group for example:

({{123, (Overscript[→, π]), 123, 231, 312, 321, 213, 132}, {(((-))^π), =, +1, +1, +1, -1, -1, -1}})

Bosonic and Fermionic States

Construct bosonic states by symmetrization

Construct fermionic states by antisymmetrization

Bosonic states (unnormalized):

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

Fermionic states (unnormalized):

(OverHat[𝒜])(Ket[α])⊗(Ket[β])=(Ket[α])⊗(Ket[β])-(Ket[β])⊗(Ket[α]), (assuming α≠β)
(OverHat[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.

the full space (of distinct particles):

𝒟=(D^N),

the bosonic subspace:

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

the fermionic subspace:

(Subscript[𝒟, F])=((D!)/(N!(D-N)!)).

Second Quantization

Fock Space

Fock States and Fock Space

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,

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,

(OverHat[𝒮])=(∑_(π∈(Subscript[S, N])))(Subscript[(OverHat[𝒫]), π]), (OverHat[𝒜])=(∑_(π∈(Subscript[S, N])))(((-))^π)(Subscript[(OverHat[𝒫]), π]),

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 property

(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])).

(BraKet[α, β])=(Subscript[δ, α β]) if (Ket[α]) and (Ket[β]) are orthonormal basis states.

The subscript ± of the insertion or deletion operators indicates whether symmetrization (+) or antisymmetrization (-) is implemented.

Boson Creation and Annihilation

Single-Mode Fock States

Prove Eq. () by definitions in Eq. () and Eq. ().

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

Using Eq. (), we can show that

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

Generic Fock States

Operator Identities

[(OverHat[A]),(OverHat[B])]=(OverHat[A])(OverHat[B])-(OverHat[B])(OverHat[A]) denotes the commutator.

The algebraic relation in Eq. () is identical to that of the creating and annihilation operators in harmonic oscillator, therefore, the elementary excitations of harmonic oscillator are indeed bosons.

Fermion Creation and Annihilation

Single-Mode Fock States

Prove Eq. () by definitions in Eq. () and Eq. ().

(Subscript[(OverHat[c]), α]^†)(Ket[(Subscript[0, α])])=(Ket[α])(SubMinus[⊳])\[ScriptOne]=(Ket[α])=(Ket[(Subscript[1, α])]),
(Subscript[(OverHat[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[(OverHat[c]), α])(Ket[(Subscript[0, α])])=0,
(Subscript[(OverHat[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

{(OverHat[A]),(OverHat[B])}=(OverHat[A])(OverHat[B])+(OverHat[B])(OverHat[A]) denotes the anti-commutator.

Quantum Statistical Physics

General Principles

Connecting Micro and Macro

At the microscopic level: physical systems are described by quantum mechanics in terms of a Hamiltonian operator (OverHat[H])

(Subscript[E, k]) - the possible energy that the system can take,

(Ket[(Subscript[E, k])]) - the corresponding quantum state of the system,

k - an index that labels the eigenstates.

At the macroscopic level: we are interested in the expectation value of physical observables (OverHat[O]),

(Bra[(Subscript[E, k])])(OverHat[O])(Ket[(Subscript[E, k])]) - the expectation value of (OverHat[O]) when the system is in the particular state (Ket[(Subscript[E, k])]) with energy (Subscript[E, k]).

(Subscript[p, k]) - the probability for the system to be in the kth eigenstate (Ket[(Subscript[E, k])]) (of energy (Subscript[E, k])) in the thermal ensemble.

The ensemble is a classical probabilistic mixture of quantum pure states (Ket[(Subscript[E, k])]) (not a quantum superposition of them), called a mixed state ensemble.

A mixed state ensemble can be specified by a set of pure state basis (Ket[(Subscript[E, k])]) together with a probability distribution (Subscript[p, k]).

Principle of Maximum Entropy

⟨H⟩=(∑_k)(Bra[(Subscript[E, k])])(OverHat[H])(Ket[(Subscript[E, k])])(Subscript[p, k])=(∑_k)(Subscript[E, k])(Subscript[p, k])=E.

(max_p)S[p]=-(∑_k)(Subscript[p, k])ln (Subscript[p, k]),
subject to:
(∑_k)(Subscript[p, k])=1,
(∑_k)(Subscript[E, k])(Subscript[p, k])=E.

Solve the constrained optimization problem Eq. () to show Eq. ().

ℒ[p]=S[p]-α(∑_k)(Subscript[p, k])-β(∑_k)(Subscript[E, k])(Subscript[p, k])
=(∑_k)(-ln (Subscript[p, k])-α-β (Subscript[E, k]))(Subscript[p, k])

∀k:((δℒ[p])/(δ (Subscript[p, k])))=0.

((δℒ[p])/(δ (Subscript[p, k])))=-ln (Subscript[p, k])-α-β (Subscript[E, k])-1,

∀k:(Subscript[p, k])=(1/(e^(α+1)))(e^(-β (Subscript[E, k]))).

The probability for the system to stay in a lower energy level is exponentially higher.

β=1/(Subscript[k, B])T is the inverse of the temperature T (and (Subscript[k, B]) is the Boltzmann constant). It will be adjusted to meet the average energy condition.

Z is the normalization coefficient for the probability distribution, also called the partition function.

Bose-Einstein Statistics

Single-Mode Problem

Eigensystem: eigenstates are labeled by (Subscript[n, α])=0,1,2,…,

(OverHat[H])(Ket[(Subscript[n, α])])=(Subscript[ϵ, α])(Subscript[n, α])(Ket[(Subscript[n, α])]),

with eigen energies

(Subscript[E, (Subscript[n, α])])=(Subscript[ϵ, α])(Subscript[n, α]).

(Subscript[p, (Subscript[n, α])])=(1/Z)(e^(-β (Subscript[E, (Subscript[n, α])])))=(1/Z)(e^(-β (Subscript[ϵ, α])(Subscript[n, α]))),

Z=(∑_((Subscript[n, α])=0)^∞)(e^(-β (Subscript[ϵ, α])(Subscript[n, α])))=(1/(1-(e^(-β (Subscript[ϵ, α]))))).

Evaluate the summation in Eq. ().

Sum[Exp[-β ϵ n],{n,0,∞}]

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Evaluate the summation in Eq. ().

Sum[n(1-Exp[-β ϵ])Exp[-β ϵ n],{n,0,∞}]

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Multi-Mode Generalization

α=1,2,…,D is the mode index, labeling single-particle states in the system.

Many-body states are labeled by a sequence of occupation numbers

n=(Subscript[n, 1]),(Subscript[n, 2]),…,(Subscript[n, D]),

where (Subscript[n, α])=0,1,2,… for bosons.

Eigensystem:

(OverHat[H])(Ket[n])=(Subscript[E, n])(Ket[n]),

with eigen energies

(Subscript[E, n])=(∑_α)(Subscript[ϵ, α])(Subscript[n, α]).

(Subscript[p, n])∝(e^(-β (Subscript[E, n])))=exp(-β(∑_α)(Subscript[ϵ, α])(Subscript[n, α]))=(∏_α) (e^(-β (Subscript[ϵ, α])(Subscript[n, α]))),

The Bose-Einstein distribution, see Eq. (),

The average total boson number

⟨N⟩=(∑_α)⟨(Subscript[n, α])⟩=(∑_α)(1/((e^( β (Subscript[ϵ, α])))-1)).

The average total energy

⟨H⟩=(∑_α)(Subscript[ϵ, α])⟨(Subscript[n, α])⟩=(∑_α)((Subscript[ϵ, α])/((e^( β (Subscript[ϵ, α])))-1)).

Fermi-Dirac Statistics

Single-Mode Problem

Eigensystem: eigenstates are labeled by (Subscript[n, α])=0,1 (Pauli exclusion principle forbid (Subscript[n, α]) to go greater than 1 for fermions),

(OverHat[H])(Ket[(Subscript[n, α])])=(Subscript[ϵ, α])(Subscript[n, α])(Ket[(Subscript[n, α])]),

with eigen energies

(Subscript[E, (Subscript[n, α])])=(Subscript[ϵ, α])(Subscript[n, α]).

(Subscript[p, (Subscript[n, α])])=(1/Z)(e^(-β (Subscript[E, (Subscript[n, α])])))=(1/Z)(e^(-β (Subscript[ϵ, α])(Subscript[n, α]))),

Z=(∑_((Subscript[n, α])=0,1))(e^(-β (Subscript[ϵ, α])(Subscript[n, α])))=1+(e^(-β (Subscript[ϵ, α]))).

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Manipulate[Block[{β=1/T},Plot[1/(Exp[β ϵ]+1),{ϵ,-5,5},PlotStyle->Red,PlotRange->{0,1},Filling->Axis,PlotRangePadding->Scaled[.05],
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Multi-Mode Generalization

α=1,2,…,D is the mode index, labeling single-particle states in the system.

Many-body states are labeled by a sequence of occupation numbers

n=(Subscript[n, 1]),(Subscript[n, 2]),…,(Subscript[n, D]),

where (Subscript[n, α])=0,1 for fermions.

Eigensystem:

(OverHat[H])(Ket[n])=(Subscript[E, n])(Ket[n]),

with eigen energies

(Subscript[E, n])=(∑_α)(Subscript[ϵ, α])(Subscript[n, α]).

(Subscript[p, n])∝(e^(-β (Subscript[E, n])))=exp(-β(∑_α)(Subscript[ϵ, α])(Subscript[n, α]))=(∏_α) (e^(-β (Subscript[ϵ, α])(Subscript[n, α]))),

The Fermi-Dirac distribution, see Eq. (),

The average total fermion number

⟨N⟩=(∑_α)⟨(Subscript[n, α])⟩=(∑_α)(1/((e^( β (Subscript[ϵ, α])))+1)).

The average total energy

⟨H⟩=(∑_α)(Subscript[ϵ, α])⟨(Subscript[n, α])⟩=(∑_α)((Subscript[ϵ, α])/((e^( β (Subscript[ϵ, α])))+1)).