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Quantum Mechanics

Field Operators

A nonrelativistic field operator is an operator-valued distribution that annihilates or creates a particle at a position, in the same sense that a delta function is a distribution rather than an ordinary function. The annihilation field is usually written ψ(x)\psi(\mathbf x), and the creation field is its adjoint ψ†(x)\psi^\dagger(\mathbf x).

The basic interpretation is:

  • ψ(x)\psi(\mathbf x) removes a particle near position x\mathbf x;
  • ψ†(x)\psi^\dagger(\mathbf x) creates a particle near position x\mathbf x;
  • products such as ψ†(x)ψ(x)\psi^\dagger(\mathbf x)\psi(\mathbf x) describe local densities;
  • integrals of field-operator products give ordinary many-particle operators.

The word “near” matters. The mathematically controlled operators are smeared with wavepackets:

df=∫d3x f∗(x)ψ(x),df†=∫d3x f(x)ψ†(x).d_f = \int d^3x\, f^*(\mathbf x)\psi(\mathbf x), \qquad d_f^\dagger = \int d^3x\, f(\mathbf x)\psi^\dagger(\mathbf x).

For normalized ff, df†∣0⟩d_f^\dagger\lvert0\rangle is the one-particle state with wavefunction f(x)f(\mathbf x). The unsmeared ψ(x)\psi(\mathbf x) is a useful distributional notation, not a bounded operator at an exact point.

Helpful background. Mode Expansions derives field completeness and smearing; Number Operators supplies the sector interpretation.

Fields acting on particle-number sectors

Section titled “Fields acting on particle-number sectors”

Choose a one-particle basis {φi(x)}\{\varphi_i(\mathbf x)\} and mode operators di,di†d_i,d_i^\dagger. The mode expansion is

ψ(x)=∑iφi(x)di,ψ†(x)=∑iφi∗(x)di†.\psi(\mathbf x) = \sum_i \varphi_i(\mathbf x)d_i, \qquad \psi^\dagger(\mathbf x) = \sum_i \varphi_i^*(\mathbf x)d_i^\dagger.

This is the coordinate-space version of creation and annihilation in modes. The field ψ(x)\psi(\mathbf x) is not a wavefunction. A wavefunction is a complex amplitude for a state; a field operator acts on Fock space and changes particle number by one.

For a normalized symmetric or antisymmetric NN-particle wavefunction ΨN(x1,…,xN)\Psi_N(\mathbf x_1,\ldots,\mathbf x_N), use the convention in which the annihilated coordinate occupies the first slot:

(ψ(x)ΨN)(x1,…,xN−1)=N ΨN(x,x1,…,xN−1).(\psi(\mathbf x)\Psi_N)(\mathbf x_1,\ldots,\mathbf x_{N-1}) = \sqrt N\, \Psi_N(\mathbf x,\mathbf x_1,\ldots,\mathbf x_{N-1}).

The same formula holds for both statistics with this convention. Moving the removed coordinate to another slot supplies the corresponding fermionic sign. The N\sqrt N factor makes the integrated removal norm equal to NN, as a count of available particles should.

In this convention the coordinate amplitude is recovered by

ΨN(x1,…,xN)=1N!⟨0∣ψ(xN)⋯ψ(x1)∣ΨN⟩.\Psi_N(\mathbf x_1,\ldots,\mathbf x_N) =\frac{1}{\sqrt{N!}}\langle0|\psi(\mathbf x_N)\cdots\psi(\mathbf x_1)|\Psi_N\rangle.

By contrast, ⟨ΨN∣ψ(x)∣ΨN⟩=0\langle\Psi_N|\psi(\mathbf x)|\Psi_N\rangle=0 in a fixed-number state, since the field connects different sectors. A nonzero density or Bose condensate does not require a nonzero field expectation in an exact number-conserving description.

Bosonic Field Algebra

Section titled “Bosonic Field Algebra”

For bosonic fields, the equal-time canonical commutation relations are

[ψ(x),ψ†(y)]=δ(3)(x−y),[\psi(\mathbf x),\psi^\dagger(\mathbf y)] = \delta^{(3)}(\mathbf x-\mathbf y),

and

[ψ(x),ψ(y)]=0,[ψ†(x),ψ†(y)]=0.[\psi(\mathbf x),\psi(\mathbf y)] = 0, \qquad [\psi^\dagger(\mathbf x),\psi^\dagger(\mathbf y)] = 0.

These equations are distributional. Smearing them with test functions gives the ordinary mode relation

[df,dg†]=⟨f∣g⟩.[d_f,d_g^\dagger] = \langle f\vert g\rangle.

For f=φif=\varphi_i and g=φjg=\varphi_j from an orthonormal basis, this reduces to [di,dj†]=δij[d_i,d_j^\dagger]=\delta_{ij}.

Fermionic Field Algebra

Section titled “Fermionic Field Algebra”

For fermionic fields, commutators are replaced by anticommutators:

{ψ(x),ψ†(y)}=δ(3)(x−y),\{\psi(\mathbf x),\psi^\dagger(\mathbf y)\} = \delta^{(3)}(\mathbf x-\mathbf y),

and

{ψ(x),ψ(y)}=0,{ψ†(x),ψ†(y)}=0.\{\psi(\mathbf x),\psi(\mathbf y)\} = 0, \qquad \{\psi^\dagger(\mathbf x),\psi^\dagger(\mathbf y)\} = 0.

Smearing gives

{df,dg†}=⟨f∣g⟩.\{d_f,d_g^\dagger\} = \langle f\vert g\rangle.

The anticommutator relation implies Pauli exclusion for complete one-particle modes. It does not mean that two fermions cannot be at nearby positions in a physical wavepacket sense; it means that the same complete spin-orbital cannot be occupied twice.

Spin and Internal Labels

Section titled “Spin and Internal Labels”

For spinful particles, the field has components:

ψs(x),ψs†(x),\psi_s(\mathbf x), \qquad \psi_s^\dagger(\mathbf x),

where ss labels spin or another discrete internal state. The fermionic equal-time algebra is

{ψs(x),ψt†(y)}=δstδ(3)(x−y),\{\psi_s(\mathbf x),\psi_t^\dagger(\mathbf y)\} = \delta_{st}\delta^{(3)}(\mathbf x-\mathbf y),

with all other same-type anticommutators equal to zero. For spinful bosons, the same formula uses commutators instead.

The complete mode label includes both spatial and internal information. For example, an electron field component ψ↑(x)\psi_\uparrow(\mathbf x) annihilates an electron at position x\mathbf x with spin-up along the chosen quantization axis. Changing the spin basis rotates the field components, but it does not change the underlying Fock-space state.

Density and Total Number

Section titled “Density and Total Number”

The local number-density operator is

n(x)=ψ†(x)ψ(x).n(\mathbf x) = \psi^\dagger(\mathbf x)\psi(\mathbf x).

For spinful particles,

n(x)=∑sψs†(x)ψs(x).n(\mathbf x) = \sum_s \psi_s^\dagger(\mathbf x)\psi_s(\mathbf x).

The total number operator is the integral of the density:

N=∫d3x n(x).N = \int d^3x\, n(\mathbf x).

Using the field algebra, one obtains

[N,ψ†(x)]=ψ†(x),[N,ψ(x)]=−ψ(x).[N,\psi^\dagger(\mathbf x)] = \psi^\dagger(\mathbf x), \qquad [N,\psi(\mathbf x)] = -\psi(\mathbf x).

These commutators say that ψ†\psi^\dagger raises total particle number by one and ψ\psi lowers it by one. They are the field-language version of the number-operator identities for mode creation and annihilation operators.

The density also satisfies

[n(x),ψ†(y)]=δ(3)(x−y)ψ†(y),[n(\mathbf x),\psi^\dagger(\mathbf y)] = \delta^{(3)}(\mathbf x-\mathbf y)\psi^\dagger(\mathbf y),

as a distributional identity. It says that the local density is raised at the point where a particle is created.

One-Body Operators in Field Language

Section titled “One-Body Operators in Field Language”

A one-body operator with position-space kernel A(x,y)A(\mathbf x,\mathbf y) is written

A^=∫d3x d3y ψ†(x)A(x,y)ψ(y).\widehat A = \int d^3x\,d^3y\, \psi^\dagger(\mathbf x) A(\mathbf x,\mathbf y) \psi(\mathbf y).

If the one-particle operator is local or differential, one often writes

A^=∫d3x ψ†(x)Axψ(x),\widehat A = \int d^3x\, \psi^\dagger(\mathbf x) A_{\mathbf x} \psi(\mathbf x),

where AxA_{\mathbf x} acts on the coordinate dependence to its right, with the usual domain and boundary-condition assumptions.

For a nonrelativistic particle in an external potential,

H^1=∫d3x ψ†(x)(−ℏ22m∇2+U(x))ψ(x).\widehat H_1 = \int d^3x\, \psi^\dagger(\mathbf x) \left( -\frac{\hbar^2}{2m}\nabla^2 +U(\mathbf x) \right) \psi(\mathbf x).

This formula is the field-language form of the one-body operator ∑ijhijdi†dj\sum_{ij}h_{ij}d_i^\dagger d_j. Combining it with two-body terms gives the standard many-particle Hamiltonian.

Pair density and interactions

Section titled “Pair density and interactions”

For a scalar species with exchange-symmetric potential v(x,y)v(\mathbf x,\mathbf y),

V^=12∫d3x d3y v(x,y)ψ†(x)ψ†(y)ψ(y)ψ(x).\widehat V=\frac12\int d^3x\,d^3y\, v(\mathbf x,\mathbf y)\psi^\dagger(\mathbf x)\psi^\dagger(\mathbf y) \psi(\mathbf y)\psi(\mathbf x).

The normally ordered density counts distinct particles; the factor 1/21/2 converts ordered to unordered pairs. It is not simply n(x)n(y)n(\mathbf x)n(\mathbf y), which also contains a self-counting term:

n(x)n(y)=ψ†(x)ψ†(y)ψ(y)ψ(x)+δ(3)(x−y)n(x).n(\mathbf x)n(\mathbf y) =\psi^\dagger(\mathbf x)\psi^\dagger(\mathbf y)\psi(\mathbf y)\psi(\mathbf x) +\delta^{(3)}(\mathbf x-\mathbf y)n(\mathbf x).

This distributional identity holds for either statistics; use smeared densities or a regulator for coincident-point products. Summing internal labels gives the spinful form. Two-Body Operators derives the index conventions and contact, Coulomb, and Hubbard examples.

Continuity Equation Preview

Section titled “Continuity Equation Preview”

For the Hamiltonian

H^=∫d3x ψ†(x)(−ℏ22m∇2+U(x))ψ(x),\widehat H = \int d^3x\, \psi^\dagger(\mathbf x) \left( -\frac{\hbar^2}{2m}\nabla^2 +U(\mathbf x) \right) \psi(\mathbf x),

the density obeys a continuity equation

∂n∂t+∇⋅j=0,\frac{\partial n}{\partial t} +\nabla\cdot\mathbf j = 0,

with current

j=ℏ2mi(ψ†∇ψ−(∇ψ†)ψ).\mathbf j = \frac{\hbar}{2mi} \left( \psi^\dagger\nabla\psi - (\nabla\psi^\dagger)\psi \right).

This is a particle-number continuity equation. Its expectation integrates to the particle count, not generally to one; unitary evolution still preserves total state probability even when a drive changes particle number. The displayed current assumes the standard kinetic term and no vector potential or spin–orbit coupling. Local density–density interactions also preserve this form, while nonlocal kinetic terms or number-changing sources require the appropriate modifications.

Nonrelativistic and relativistic fields

Section titled “Nonrelativistic and relativistic fields”

These are already quantum fields, with nonrelativistic dynamics. Equal-time field algebra alone does not impose Lorentz symmetry or relativistic microcausality. The bridge to relativistic QFT explains the extra spacetime structure and why a local relativistic field generally contains both creation and annihilation parts.

Exercises

Section titled “Exercises”
  1. Smeared field algebra. For bosonic fields, show that [df,dg†]=⟨f∣g⟩[d_f,d_g^\dagger]=\langle f\vert g\rangle when
df=∫d3x f∗(x)ψ(x).d_f = \int d^3x\, f^*(\mathbf x)\psi(\mathbf x).
Solution

Use the field commutator:

[df,dg†]=∫d3x d3y f∗(x)g(y)[ψ(x),ψ†(y)]=∫d3x d3y f∗(x)g(y)δ(3)(x−y)=∫d3x f∗(x)g(x)=⟨f∣g⟩.\begin{aligned} [d_f,d_g^\dagger] &= \int d^3x\,d^3y\, f^*(\mathbf x)g(\mathbf y) [\psi(\mathbf x),\psi^\dagger(\mathbf y)] \\ &= \int d^3x\,d^3y\, f^*(\mathbf x)g(\mathbf y) \delta^{(3)}(\mathbf x-\mathbf y) \\ &= \int d^3x\, f^*(\mathbf x)g(\mathbf x) = \langle f\vert g\rangle. \end{aligned}
  1. Particle-number commutator. Starting from N=∫d3x ψ†(x)ψ(x)N=\int d^3x\,\psi^\dagger(\mathbf x)\psi(\mathbf x), explain why [N,ψ†(y)]=ψ†(y)[N,\psi^\dagger(\mathbf y)]=\psi^\dagger(\mathbf y).
Solution

For bosons, use [AB,C]=A[B,C]+[A,C]B[AB,C]=A[B,C]+[A,C]B:

[N,ψ†(y)]=∫d3x [ψ†(x)ψ(x),ψ†(y)]=∫d3x ψ†(x)[ψ(x),ψ†(y)]=∫d3x ψ†(x)δ(3)(x−y)=ψ†(y).\begin{aligned} [N,\psi^\dagger(\mathbf y)] &= \int d^3x\, [\psi^\dagger(\mathbf x)\psi(\mathbf x),\psi^\dagger(\mathbf y)] \\ &= \int d^3x\, \psi^\dagger(\mathbf x) [\psi(\mathbf x),\psi^\dagger(\mathbf y)] \\ &= \int d^3x\, \psi^\dagger(\mathbf x) \delta^{(3)}(\mathbf x-\mathbf y) \\ &= \psi^\dagger(\mathbf y). \end{aligned}

The fermionic result is the same for the ordinary commutator with the even operator NN, although the intermediate algebra uses anticommutation relations.

  1. Density of a normalized N-particle state. Use the field action to show that ⟨n(x)⟩=N∫d3x2⋯d3xN ∣ΨN(x,x2,…)∣2\langle n(\mathbf x)\rangle=N\int d^3x_2\cdots d^3x_N\,|\Psi_N(\mathbf x,\mathbf x_2,\ldots)|^2. What is its spatial integral?
Solution

The density expectation is the norm squared of ψ(x)∣ΨN⟩\psi(\mathbf x)|\Psi_N\rangle. Inserting the field action gives the factor N and the integral over the remaining coordinates. Integrating also over x\mathbf x gives N by normalization of ΨN\Psi_N. Thus ⟨n(x)⟩/N\langle n(\mathbf x)\rangle/N is the normalized one-particle position density when N>0N>0.

References

Section titled “References”
  • A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
  • A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
  • J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley, 1988.
  • M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
  • L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Superfluidity, Oxford University Press, 2016.
  • R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.