Introduction to Second Quantization

Introduction to Second Quantization

A particle in a box

The Infinite Potential Box

It’s commonly said that a Quantum Field is a bunch of Harmonic Oscillators. To see what’s really going on, let’s start with a particle in a box:

V(x)={0,0<x<L, otherwise V(x)=\left\{\begin{array}{ll}{0,} & {0<x<L} \\ {\infty,} & {\text { otherwise }}\end{array}\right.

Inside the box, the Schrodinger equation is:

itψ(x,t)=22m2x2ψ(x,t)+V(x)ψ(x,t)i \hbar \frac{\partial}{\partial t} \psi(x, t)=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} \psi(x, t)+V(x) \psi(x, t)

For a box of size LL, we have the following boundary conditions:

ψ(0)=ψ(L)=0\psi(0)=\psi(L)=0

To obtain ψ(x)\psi(x), just guess an exponential. The solution is:

ψn(x)=2Lsin(knx)=2Lsin(nπxL)\begin{aligned} \psi_{n}(x)&=\sqrt{\frac{2}{L}} \sin \left(k_n x\right)\\ &=\sqrt{\frac{2}{L}} \sin \left(\frac{n \pi x}{L}\right) \end{aligned}

where

kn=nπLk_n=\frac{n \pi}{L}

Using the De Broglie relation, we note that momentum is now quantized:

pn=kn=nπLp_n=\hbar k_n = \frac{n \pi \hbar}{L}

the energy is, for n=0,±1,±2,n=0, \pm 1, \pm 2, \ldots

En=2π2n22mL2=pn22mE_{n}=\frac{\hbar^{2} \pi^{2} n^{2}}{2 m L^{2}} = \frac{p_n^{2}}{2 m}

Periodic Boundary Box

However, the momentum is not a good quantum number since it does not commute with the Hamiltonian. due to the V(x)V(x) term. This causes some technical issues since our momentum states aren’t eigenstates. We can simply just get rid of the potential V(x)V(x) and impose periodic boundary conditions. The new Schrodinger equation is:

itψ(x,t)=22m2x2ψ(x,t)i \hbar \frac{\partial}{\partial t} \psi(x, t)=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} \psi(x, t)

With periodic boundary conditions:

ψ(0)=ψ(L),ψ(0)=ψ(L)\psi(0)=\psi(L), \quad \psi^{\prime}(0)=\psi^{\prime}(L)

which imply

ψ(x+L)=ψ(x)\psi(x+L)=\psi(x)

And so the solution is

ψn(x)=eiknxL\psi_{n}(x)=\frac{e^{i k_{n} x}}{\sqrt{L}}

where

kn=2πnL,n=0,±1,±2,k_{n}=\frac{2 \pi n}{L}, \quad n=0, \pm 1, \pm 2, \ldots

And the energy is, for n=0,±1,±2,n=0, \pm 1, \pm 2, \ldots

En=2π2n22mL2=pn22mE_{n}=\frac{\hbar^{2} \pi^{2} n^{2}}{2 m L^{2}} = \frac{p_n^{2}}{2 m}

Now we can see that momentum is now an eigenstate:

p^ψn(x)=ixψn(x)=pnψn(x)pn=kn=2πnL\begin{aligned} \hat{p} \psi_{n}(x)&=-i \frac{\partial}{\partial x} \psi_{n}(x)=p_{n} \psi_{n}(x) \\ p_{n}&=\hbar k_{n}=\frac{2 \pi \hbar n}{L} \end{aligned}

The results are the same as before, except now our new momentum pnp_n is double of the old momentum, and the new momentum states are now eigenstates.

Two particles in a box

Let’s now investigate a non interacting multi particle system. Let’s also rename the letter npmn \to p_m, so we can free up the letter nn for something new.

Suppose the first particle is in state p1|p_1\rangle, and the second particle is in state p2|p_2 \rangle. Then as usual, we describe the two particle system with:

p1p2|p_1 p_2 \rangle

Apply the momentum and energy operators we get:

p^p1p2=(p1+p2)p1p2\hat{p}\left|p_{1} p_{2}\right\rangle=\left(p_{1}+p_{2}\right)\left|p_{1} p_{2}\right\rangleH^p1p2=(Ep1+Ep2)p1p2\hat{H}\left|p_{1} p_{2}\right\rangle=\left(E_{p_{1}}+E_{p_{2}}\right)\left|p_{1} p_{2}\right\rangle

and we can write the total energy is given by:

mnpmEpm\sum_m n_{p_m} E_{p_m}

where npmn_{p_m} is the total number of particles in the state pm|p_m\rangle.

Changing the Notation

In single particle quantum mechanics we might write:

ABC|ABC\rangle

where

  • AA is the state of the first particle,
  • BB is the state of the second particle, and
  • CC is the state of the third particle.

So if

  • A=p1A = p_1
  • B=p2B = p_2
  • C=p1C = p_1

Then we would write

ABC=p1p2p1|{ABC}\rangle=\left|p_{1} p_{2} p_{1}\right\rangle

This would make sense if particles were distinguishable, but they are not. The laws of indistinguishable particles are:

  • There are only two types of particles in the Universe: fermions and bosons.

  • Looking at the wave functions in the position basis:

  • We can define the exchange operator:

    P^2AB=P^BA=AB \hat{P}^2 |AB\rangle =\hat{P}|BA\rangle = |AB\rangle

    Since P^2=1\hat{P}^2 = 1, this leaves:

    P^AB=λBA\hat{P}\left|AB\right\rangle=\lambda\left|BA\right\rangle

    We see that λ2=1\lambda^2 = 1 and so λ=±1\lambda = \pm 1.

    • If you swap two identical bosons nothing happens: AB=BA|AB\rangle = |BA\rangle. This is λ=1\lambda = 1.
    • If you swap two identical fermions you get a minus sign: AB=BA|AB\rangle = -|BA\rangle. This is λ=1\lambda = -1 and is known as the Pauli exclusion principle.

We can make a change of notation to accommodate this. This new notation is called Second quantization, and the old notation is called First quantization.

Before in first quantization, we would have said: Which particle is in which state? But according to the rules above, this is nonsense. In second quantization, we ask: How many particles in each state?

We now list the number of identical particles in each quantum state:

ParticlesOld notationNew notation
00\|0\rangle00\|00\rangle
1q1\|q_1\rangle10\|10\rangle
1q2\|q_2\rangle01\|01\rangle
2q1q1\|q_1q_1\rangle20\|20\rangle
2q1q2\|q_1q_2\rangle11\|11\rangle
2q2q2\|q_2q_2\rangle02\|02\rangle
3q1q1q1\|q_1q_1q_1\rangle30\|30\rangle

So, now when we apply the Hamiltonian, we get:

H^n1n2n3=[mnpmEpm]n1n2n3\hat{H}\left|n_{1} n_{2} n_{3} \ldots\right\rangle=\left[\sum_{m} n_{p_{m}} E_{p_{m}}\right]\left|n_{1} n_{2} n_{3} \ldots\right\rangle

The computation is now very natural in second quantization, we just count the number of particles and multiply by the energies:

mnpmEpm=n1Ep1+n2Ep2+\sum_{m} n_{p_{m}} E_{p_{m}}=n_1 E_{p_1}+n_2 E_{p_2}+\ldots

The Harmonic Oscillator

The quantum harmonic oscillator has energy

E=(n+12)ωE = \left(n + \frac{1}{2}\right)\hbar \omega

If we throw away the zero point energy, we can write this as:

E=nωE = n\hbar \omega

For NN independent oscillators, we have the total energy is:

k=1Nnkωk\sum_{k=1}^{N} n_{k} \hbar \omega_{k}

The total energy of the oscillator looks exactly the same as the many body system. These two systems are special in that their energy eigenvalues are linearly spaced. This is why we can associate the energy of identical particles with that of the harmonic oscillator.

So having 2ω2 \hbar \omega of quanta can mean either we have two photons with energy ω\hbar \omega, or a single photon with energy 2ω2 \hbar \omega.

Harmonic OscillatorIdentical Particles
QuantaParticles in momentum states
kkth Oscillatormmth momentum mode
k=1Nnkωk\sum_{k=1}^{N} n_{k} \hbar \omega_{k}m=1NnpmEpm\sum_{m=1}^{N} n_{p_{m}} E_{p_{m}}

Note that we’ve done this mathematically: we haven’t physically replaced identical particles with harmonic oscillators.

Raising and lowering operators in momentum space

States defined by the quantum harmonic oscillator come with the raising a^\hat{a}^{\dagger} and lowering a^\hat{a} operators, allowing us to raise and lower states in the harmonic oscillator system. Here n|n\rangle is the state of the single harmonic oscillator:

n=(a^)nn!0|n\rangle=\frac{\left(\hat{a}^{\dagger}\right)^{n}}{\sqrt{n !}}|0\rangle

The wavefunctions ψ(x)\psi(x) of the Harmonic Oscillator are quite complicated, but if you are curious about what the wave functions of the Harmonic Oscillator look like they are here:

The raising operator allows us to raise states:

a^n=n+1n+1\hat{a}^\dagger |n\rangle = \sqrt{n+1}|n+1\rangle

The lowering operator allows us to lower states:

a^n=nn1\hat{a}|n\rangle ={\sqrt {n}}|n-1\rangle

This is just like climbing a ladder! For this, these are also called ladder operators.

Image: David J. Griffths, Introduction to Quantum Mechanics pg.61, Figure 2.5

Image: David J. Griffths, Introduction to Quantum Mechanics pg.61, Figure 2.5

Since the quantum harmonic oscillator is the same as the identical particles system, we can use the raising a^\hat{a}^{\dagger} and lowering a^\hat{a}

n1n2=k1nk!(a^k)nk0\left|n_{1} n_{2} \cdots\right\rangle=\prod_{k} \frac{1}{\sqrt{n_{k}!}} \left(\hat{a}_{k}^{\dagger}\right)^{n_{k}}|0\rangle

And say that n1n2\left|n_{1} n_{2} \cdots\right\rangle is the state of the many body system! To create a particle with momentum kk, just use the raising operator a^k\hat{a}^\dagger_k!

Some physicists will tell you that particles are operators. This can be confusing, but just remember what they act on: the vacuum state 0|0\rangle.

  • In second quantization, we can create and destroy particles in a state using the raising and lowering operators of the harmonic oscillator.
  • Right now our box is finite, so we still have to take the infinite limit.
  • We’re still in momentum space so we have to Fourier transform this later.

Indistinguishability and Symmetry

Suppose that we had two states as such:

a^p10=10,a^p20=01\hat{a}_{p_{1}}^{\dagger}|0\rangle=|10\rangle, \quad \hat{a}_{p_{2}}^{\dagger}|0\rangle=|01\rangle

Let’s add one more particle:

a^p2a^p1011,a^p1a^p2011\hat{a}_{p_{2}}^{\dagger} \hat{a}_{p_{1}}^{\dagger}|0\rangle \propto|11\rangle, \quad \hat{a}_{p_{1}}^{\dagger} \hat{a}_{p_{2}}^{\dagger}|0\rangle \propto|11\rangle

Both expressions are proportional to the same state, so they must be proportional:

a^p1a^p2=λa^p2a^p1\hat{a}_{p_{1}}^{\dagger} \hat{a}_{p_{2}}^{\dagger}=\lambda \hat{a}_{p_{2}}^{\dagger} \hat{a}_{p_{1}}^{\dagger}

We said before that λ=±1\lambda = \pm 1 depending on whether the particles are fermions or bosons, so let’s take a look at each case:

Bosons

When λ=1\lambda = 1, we have the particles are bosons. Therefore, we have that the operators commute

b^p2b^p1=b^p1b^p2\hat{b}_{p_{2}}^{\dagger} \hat{b}_{p_{1}}^{\dagger}=\hat{b}_{p_{1}}^{\dagger} \hat{b}_{p_{2}}^{\dagger}

and so the commutator, defined as:

[A^,B^]:=A^B^B^A^\left[\hat{A}, \hat{B}\right]:=\hat{A}\hat{B}-\hat{B}\hat{A}

is zero, since the operators commute.

[b^i,b^j]=b^ib^jb^jb^i=0\left[\hat{b}_{i}^{\dagger}, \hat{b}_{j}^{\dagger}\right]=\hat{b}_{i}^{\dagger} \hat{b}_{j}^{\dagger}-\hat{b}_{j}^{\dagger} \hat{b}_{i}^{\dagger}=0

The same is true for the lowering operators:

[b^i,b^j]=b^ib^jb^jb^i=0\left[\hat{b}_{i}, \hat{b}_{j}\right] =\hat{b}_{i} \hat{b}_{j}-\hat{b}_{j}\hat{b}_{i}=0

To make it exactly the same as the harmonic oscillator, we define:

[b^i,b^j]=δji\left[\hat{b}_{i}, \hat{b}_{j}^{\dagger}\right]=\delta^i_{j}

That is, if you raise and then lower you it is equal to the Kronecker delta:

δji={0 if ij1 if i=j\delta^i_{ j}=\left\{\begin{array}{ll}{0} & {\text { if } i \neq j} \\ {1} & {\text { if } i=j}\end{array}\right.

And so if the particles are bosons it doesn’t matter which order you put particles in the states.

b^p1b^p20=b^p2b^p10=11\hat{b}_{p_{1}}^{\dagger} \hat{b}_{p_{2}}^{\dagger}|0\rangle=\hat{b}_{p_{2}}^{\dagger} \hat{b}_{p_{1}}^{\dagger}|0\rangle=\left|1 1\right\rangle

Fermions

When λ=1\lambda = -1, the particles are fermions. The operators then anti-commute:

c^p2c^p1=c^p1c^p2\hat{c}_{p_{2}}^{\dagger} \hat{c}_{p_{1}}^{\dagger}=-\hat{c}_{p_{1}}^{\dagger} \hat{c}_{p_{2}}^{\dagger}

And so the anti-commutator, defined as

{A^,B^}:=A^B^+B^A^\left\{\hat{A}, \hat{B}\right\}:=\hat{A}\hat{B}+\hat{B}\hat{A}

is zero, since the operators anti-commute.

{c^i,c^j}=c^ic^j+c^jc^i=0\left\{\hat{c}_{i}^{\dagger}, \hat{c}_{j}^{\dagger}\right\}=\hat{c}_{i}^{\dagger} \hat{c}_{j}^{\dagger}+\hat{c}_{j}^{\dagger} \hat{c}_{i}^{\dagger}=0

The same is true for the lowering operators:

{c^i,c^j}=c^ic^j+c^jc^i=0\Big\{\hat{c}_{i}, \hat{c}_{j}\Big\} =\hat{c}_{i} \hat{c}_{j}+\hat{c}_{j}\hat{c}_{i}=0

And similarly, we define:

{c^i,c^j}=δji\left\{\hat{c}_{i}, \hat{c}_{j}^{\dagger}\right\}=\delta^i_{j}

The anti-commutation relations are the exact same as the quantum harmonic oscillator, except that we have anti-commutators instead of commutators.

Note that setting i=ji = j we get that

c^ic^i+c^ic^i=0\hat{c}_{i}^{\dagger} \hat{c}_{i}^{\dagger}+\hat{c}_{i}^{\dagger} \hat{c}_{i}^{\dagger}=0

and we get the Pauli Exclusion principle, which says that each state can only contain a single fermion.

c^ic^i=0\hat{c}_{i}^{\dagger} \hat{c}_{i}^{\dagger}=0

Lastly, we have that

c^ic^j0=c^jc^i0\hat{c}_{i}^{\dagger} \hat{c}_{j}^{\dagger}|0\rangle=-\hat{c}_{j}^{\dagger} \hat{c}_{i}^{\dagger}|0\rangle

So it matters which order you put the particles into the states!

The continuum limit

Our particles are in confined in a box, but a quantum field theory usually needs to work outside a finite box. Just take the length of the box to be infinity, and our Kronecker deltas become Dirac deltas in 1D:

δjiδ(xx0)\delta^i_{j} \rightarrow \delta(x-x_0)

Where the Dirac delta is the continuous version of the Kronecker delta:

δ(x)={+,x=00,x0\delta(x)=\left\{\begin{array}{ll}{+\infty,} & {x=0} \\ {0,} & {x \neq 0}\end{array}\right.

The Kronecker delta has the property that:

i=viδji=vj\sum_{i=-\infty}^{\infty} v_{i} \delta^i_{j}=v_{j}

The Dirac delta behaves similarly:

δ(xy)f(x)dx=f(y)\int_{-\infty}^{\infty} \delta(x-y) f(x) d x=f(y)

And so for 3D space, we have:

δ(3)(xx0):=δ(xx0)δ(yy0)δ(zz0)\delta^{(3)}(\mathbf{x} - \mathbf{x_0}) := \delta(x-x_0)\delta(y-y_0)\delta(z-z_0)

Change of basis

There’s one last technique we need which is the change of basis in second quantization. So far, our operators work only in momentum space. Here we will give a proof that will work for changing to any basis.

In first quantization we have:

ψα(x)xα\psi_{\alpha}(x) \equiv\langle x | \alpha\rangle

For both fermions and bosons we have:

α=a^α0|\alpha\rangle= \hat{a}_{\alpha}^{\dagger}|0\rangle

where a^α\hat{a}_{\alpha}^{\dagger} is the respective raising operator.

Suppose we want to change to the basis α~|\tilde{\alpha}\rangle.

ψ~α~(x)xα~\tilde{\psi}_{\tilde{\alpha}}(x) \equiv\langle x | \tilde{\alpha}\rangle

with

α~=a^α~0|\tilde{\alpha}\rangle= \hat{a}_{\tilde{\alpha}}^{\dagger}|0\rangle

We use the identity operator:

I=ααα=α~α~α~I=\sum_{\alpha}|\alpha\rangle\langle\alpha|=\sum_{\tilde{\alpha}}|\tilde{\alpha}\rangle\langle\tilde{\alpha}|

hence

a^α0=α=α~α~α~α=α~α~αα~=α~α~αa^α~0\hat{a}_{\alpha}^{\dagger}|0\rangle=|\alpha\rangle=\sum_{\tilde{\alpha}}|\tilde{\alpha}\rangle\langle\tilde{\alpha} | \alpha\rangle \\= \sum_{\tilde{\alpha}}\langle\tilde{\alpha} | \alpha\rangle|\tilde{\alpha}\rangle=\sum_{\tilde{\alpha}}\langle\tilde{\alpha} | \alpha\rangle \hat{a}_{\tilde{\alpha}}^{\dagger}|0\rangle

therefore:

a^α=α~α~αa^α~\hat{a}_{\alpha}^{\dagger}=\sum_{\tilde{\alpha}}\langle\tilde{\alpha} | \alpha\rangle \hat{a}_{\tilde{\alpha}}^{\dagger}

Taking the complex transpose/dagger of both sides:

a^α=α~αα~a^α~\hat{a}_{\alpha}=\sum_{\tilde{\alpha}}\langle\alpha | \tilde{\alpha}\rangle \hat{a}_{\tilde{\alpha}}

The inverse transformations are derived very similarly:

a^α~=ααα~a^αa^α~=αα~αa^α\begin{aligned} \hat{a}_{\tilde{\alpha}}^{\dagger} &=\sum_{\alpha}\langle\alpha | \tilde{\alpha}\rangle \hat{a}_{\alpha}^{\dagger} \\ \hat{a}_{\tilde{\alpha}} &=\sum_{\alpha}\langle\tilde{\alpha} | \alpha\rangle\hat{a}_{\alpha} \end{aligned}

The wave functions follow a very similar relation, the same as in first quantization:

ψα(x)=xα=x(α~α~αα~)=α~α~αxα~=α~α~αψ~α~(x)\psi_{\alpha}(x)=\langle x | \alpha\rangle = \langle x|\left(\sum_{\tilde{\alpha}}\langle\tilde{\alpha} | \alpha\rangle|\tilde{\alpha}\rangle\right)\\=\sum_{\tilde{\alpha}}\langle\tilde{\alpha} | \alpha\rangle\langle x | \tilde{\alpha}\rangle=\sum_{\tilde{\alpha}}\langle\tilde{\alpha} | \alpha\rangle \tilde{\psi}_{\tilde{\alpha}}(x)

and

ψ~α~(x)=ααα~ψα(x)\tilde{\psi}_{\tilde{\alpha}}(x)=\sum_{\alpha}\langle\alpha | \tilde{\alpha}\rangle \psi_{\alpha}(x)

Creation and destruction operators in position space

Finally, we have the field operators. We put a hat on them so we don’t confuse them with wavefunctions. These are the operators that create and destroy particles at a position xx. Technically physicists use “creation and destruction” interchangeably with “raising and lowering” since they are behave the same mathematically, but I think it sounds cooler and is less confusing if we give different names depending on whether we are in the position or momentum space.

ψ^(x)=ααxa^α=αψα(x)a^αψ^(x)=αxαa^α=αψα(x)a^α\begin{aligned} \hat{\psi}^{\dagger}(x) &=\sum_{\alpha}\langle\alpha | x\rangle \hat{a}_{\alpha}^{\dagger}=\sum_{\alpha} \psi_{\alpha}^{*}(x) \hat{a}_{\alpha}^{\dagger} \\ \hat{\psi}(x) &=\sum_{\alpha}\langle x | \alpha\rangle \hat{a}_{\alpha}=\sum_{\alpha} \psi_{\alpha}(x) \hat{a}_{\alpha} \end{aligned}

We can get them for free from the equations

a^α~=ααα~a^αa^α~=αα~αa^α\begin{aligned} \hat{a}_{\tilde{\alpha}}^{\dagger} &=\sum_{\alpha}\langle\alpha | \tilde{\alpha}\rangle \hat{a}_{\alpha}^{\dagger} \\ \hat{a}_{\tilde{\alpha}} &=\sum_{\alpha}\langle\tilde{\alpha} | \alpha\rangle\hat{a}_{\alpha} \end{aligned}

The field operators are the operators that create and destroy particles in the position basis. Just like in first quantization, position and momentum space are related by the Fourier transform!

For bosons, we have:

[ψ^(x),ψ^(x)]=0[ψ^(x),ψ^(x)]=0[ψ^(x),ψ^(x)]=δ(xx)\begin{aligned}\left[\hat{\psi}(x), \hat{\psi}\left(x^{\prime}\right)\right] &= 0 \\ \left[\hat{\psi}^{\dagger}(x), \hat{\psi}^{\dagger}\left(x^{\prime}\right)\right]&=0 \\ \left[\hat{\psi}(x), \hat{\psi}^{\dagger}\left(x^{\prime}\right)\right] &=\delta\left(x-x^{\prime}\right) \end{aligned}

And for fermions, we have:

{ψ^(x),ψ^(x)}=0{ψ^(x),ψ^(x)}=0{ψ^(x),ψ^(x)}=δ(xx)\begin{aligned}\left\{\hat{\psi}(x), \hat{\psi}\left(x^{\prime}\right)\right\} &= 0 \\ \left\{\hat{\psi}^{\dagger}(x), \hat{\psi}^{\dagger}\left(x^{\prime}\right)\right\}&=0 \\ \left\{\hat{\psi}(x), \hat{\psi}^{\dagger}\left(x^{\prime}\right)\right\} &=\delta\left(x-x^{\prime}\right) \end{aligned}

Summary

  • Second quantization differs from first quantization by listing the number of identical particles in each quantum state. It is mainly used to describe many-body physics.

  • We do this by raising the vacuum state 0|0 \rangle.

  • There are three commutation rules for Bosons:

    [bi,bk]=0,[b^i,b^k]=0,[bi,bj]=δ(xx)\left[b_{i}^{\dagger}, b_{k}^{\dagger}\right]=0, \quad\left[\hat{b}_{i}, \hat{b}_{k}\right]=0, \\ \left[b_{i}, b_{j}^{\dagger}\right]=\delta\left(x-x^{\prime}\right)
  • There are three anti-commutation rules for Fermions:

    {ci,ck}=0,{c^i,c^k}=0,{ci,cj}=δ(xx)\left\{c_{i}^{\dagger}, c_{k}^{\dagger}\right\}=0, \quad \Big \{ \hat{c}_{i}, \hat{c}_{k}\Big \}=0, \\ \left\{c_{i}, c_{j}^{\dagger}\right\}=\delta\left(x-x^{\prime}\right)

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Zhi Han