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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.

What the Fields Mean

Section titled “What the Fields Mean”

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 bosonic NN-particle wavefunction ΨN(x1,…,xN)\Psi_N(\mathbf x_1,\ldots,\mathbf x_N), the annihilation field acts schematically as

(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}).

For fermions, an analogous formula holds once an insertion convention is fixed; the antisymmetry of the wavefunction supplies the signs. In occupation language those signs are the same signs produced by the fermionic anticommutation relations.

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 orthonormal modes f=g=φif=g=\varphi_i, 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.

Two-Body Interactions in Field Language

Section titled “Two-Body Interactions in Field Language”

A number-conserving two-body interaction with symmetric potential v(x,y)v(\mathbf x,\mathbf y) is written

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

The factor 1/21/2 avoids double counting unordered pairs. The displayed operator ordering is part of the convention, especially for fermions.

For spinful particles, spin labels are summed. For example,

V^=12∑st∫d3x d3y ψs†(x)ψt†(y)v(x,y)ψt(y)ψs(x).\widehat V = \frac12 \sum_{st} \int d^3x\,d^3y\, \psi_s^\dagger(\mathbf x) \psi_t^\dagger(\mathbf y) v(\mathbf x,\mathbf y) \psi_t(\mathbf y) \psi_s(\mathbf x).

For a spinless bosonic contact interaction,

v(x,y)=g δ(3)(x−y),v(\mathbf x,\mathbf y) = g\,\delta^{(3)}(\mathbf x-\mathbf y),

so

V^contact=g2∫d3x ψ†(x)ψ†(x)ψ(x)ψ(x).\widehat V_{\mathrm{contact}} = \frac g2 \int d^3x\, \psi^\dagger(\mathbf x) \psi^\dagger(\mathbf x) \psi(\mathbf x) \psi(\mathbf x).

This is a standard effective low-energy form. Its coupling gg depends on the physical model and regularization; it should not be treated as a universal microscopic constant.

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 the field-operator version of probability conservation in wave mechanics. It assumes the standard kinetic term and no source or sink terms that change particle number.

Bridge to QFT

Section titled “Bridge to QFT”

Nonrelativistic field operators are already field-like because particles are excitations of modes and local densities are written with field products. But this page is still nonrelativistic. It assumes:

  • a single time parameter;
  • equal-time commutation or anticommutation relations;
  • a chosen one-particle Hilbert space;
  • fixed particle species;
  • no Lorentz-covariant microcausality condition.

Relativistic QFT changes the role of fields. Fields become local spacetime objects constrained by Lorentz symmetry, antiparticles and particle production are built into the representation theory, and locality is expressed through spacelike commutation or anticommutation conditions. Nonrelativistic field operators are the right bridge, but they are not yet the full relativistic theory.

Common Mistakes

Section titled “Common Mistakes”
  • Treating ψ(x)\psi(\mathbf x) as a wavefunction instead of an operator that changes particle number.
  • Forgetting that unsmeared field operators are distributions.
  • Using commutators for fermions or anticommutators for bosons.
  • Dropping spin or internal labels from complete field components.
  • Missing the factor 1/21/2 in symmetric two-body interactions.
  • Assuming the contact interaction is fundamental without specifying the effective model.
  • Treating nonrelativistic equal-time field algebra as already equivalent to relativistic QFT.
Section titled “Cross-Links”

References

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

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. Number from density. Use ψ(x)=∑iφi(x)di\psi(\mathbf x)=\sum_i\varphi_i(\mathbf x)d_i to show that N=∫d3x ψ†(x)ψ(x)N=\int d^3x\,\psi^\dagger(\mathbf x)\psi(\mathbf x) equals ∑idi†di\sum_i d_i^\dagger d_i.
Solution

Substitute the mode expansion:

N=∫d3x∑ijφi∗(x)φj(x)di†dj=∑ij(∫d3x φi∗(x)φj(x))di†dj=∑idi†di.\begin{aligned} N &= \int d^3x \sum_{ij} \varphi_i^*(\mathbf x) \varphi_j(\mathbf x) d_i^\dagger d_j \\ &= \sum_{ij} \left( \int d^3x\, \varphi_i^*(\mathbf x)\varphi_j(\mathbf x) \right) d_i^\dagger d_j \\ &= \sum_i d_i^\dagger d_i. \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. One-body potential. Write the field-operator form of a one-particle potential U(x)U(\mathbf x) and identify the density.
Solution

The potential is multiplication by U(x)U(\mathbf x), so

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

Since U(x)U(\mathbf x) is a scalar function, this can be written

U^=∫d3x U(x)n(x),n(x)=ψ†(x)ψ(x).\widehat U = \int d^3x\, U(\mathbf x)n(\mathbf x), \qquad n(\mathbf x) = \psi^\dagger(\mathbf x)\psi(\mathbf x).
  1. Contact interaction. Derive the spinless bosonic contact form from v(x,y)=gδ(3)(x−y)v(\mathbf x,\mathbf y)=g\delta^{(3)}(\mathbf x-\mathbf y).
Solution

Start with

V^=12∫d3x d3y ψ†(x)ψ†(y)gδ(3)(x−y)ψ(y)ψ(x).\widehat V = \frac12 \int d^3x\,d^3y\, \psi^\dagger(\mathbf x) \psi^\dagger(\mathbf y) g\delta^{(3)}(\mathbf x-\mathbf y) \psi(\mathbf y) \psi(\mathbf x).

Use the delta function to perform the y\mathbf y integral:

V^=g2∫d3x ψ†(x)ψ†(x)ψ(x)ψ(x).\widehat V = \frac g2 \int d^3x\, \psi^\dagger(\mathbf x) \psi^\dagger(\mathbf x) \psi(\mathbf x) \psi(\mathbf x).