10.2 Creation and annihilation operators

Creation and annihilation operators are key tools in quantum mechanics for describing multi-particle systems. They allow us to add or remove particles from quantum states, making it easier to work with complex systems of many particles.

These operators are crucial for understanding second quantization, which is a powerful method for dealing with systems of identical particles. By using creation and annihilation operators, we can build up multi-particle states and calculate important physical quantities.

Creation and annihilation operators

Definition and action on Fock states

• Creation operators ($a^{\dagger}$) increase the number of particles in a specific quantum state by one
  • The action of the creation operator on a Fock state $|n\rangle$ is given by $a^{\dagger}|n\rangle = \sqrt{n+1}|n+1\rangle$, where $n$ is the number of particles in the state
  • Example: $a^{\dagger}|2\rangle = \sqrt{3}|3\rangle$, adding one particle to the state with 2 particles
• Annihilation operators ($a$) decrease the number of particles in a specific quantum state by one
  • The action of the annihilation operator on a Fock state $|n\rangle$ is given by $a|n\rangle = \sqrt{n}|n-1\rangle$, where $n$ is the number of particles in the state
  • Example: $a|3\rangle = \sqrt{3}|2\rangle$, removing one particle from the state with 3 particles
• The annihilation operator acting on the vacuum state ($|0\rangle$) gives zero: $a|0\rangle = 0$
  • The vacuum state represents the state with no particles

Number operator

• The number operator ($N$) is defined as $N = a^{\dagger}a$
  • Its eigenvalue for a Fock state $|n\rangle$ is $n$, the number of particles in the state
  • Example: $N|3\rangle = 3|3\rangle$, indicating that the state has 3 particles
• The number operator can be used to determine the number of particles in a given state
  • It is useful for calculating expectation values and probabilities related to particle numbers

Bosonic vs fermionic operators

Commutation relations for bosonic operators

• Bosonic creation and annihilation operators satisfy the commutation relations:
  • $[a, a^{\dagger}] = aa^{\dagger} - a^{\dagger}a = 1$
  • $[a, a] = [a^{\dagger}, a^{\dagger}] = 0$
• These commutation relations are crucial for determining the properties of bosonic systems
  • They allow for the construction of multi-particle states and the calculation of expectation values
• Example: Photons and phonons are bosonic particles that obey these commutation relations

Anti-commutation relations for fermionic operators

• Fermionic creation and annihilation operators satisfy the anti-commutation relations:
  • $\{a, a^{\dagger}\} = aa^{\dagger} + a^{\dagger}a = 1$
  • $\{a, a\} = \{a^{\dagger}, a^{\dagger}\} = 0$
• The anti-commutation relations are essential for describing the properties of fermionic systems
  • They lead to the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state
• Example: Electrons and quarks are fermionic particles that obey these anti-commutation relations

Multi-particle states

Constructing multi-particle states using creation operators

• Multi-particle states can be constructed by applying creation operators to the vacuum state ($|0\rangle$)
• For bosons, a multi-particle state with $n$ particles in a specific quantum state $|\psi\rangle$ is given by:
  • $|n\rangle = \frac{(a^{\dagger})^n}{\sqrt{n!}}|0\rangle$, where $(a^{\dagger})^n$ represents the creation operator applied $n$ times
  • The normalization factor $\frac{1}{\sqrt{n!}}$ ensures that the multi-particle state is properly normalized
  • Example: $|3\rangle = \frac{(a^{\dagger})^3}{\sqrt{3!}}|0\rangle$ represents a state with 3 bosons in the same quantum state
• For fermions, a multi-particle state with particles in different quantum states $|\psi_1\rangle, |\psi_2\rangle, \ldots, |\psi_n\rangle$ is given by:
  • $|\psi_1, \psi_2, \ldots, \psi_n\rangle = a^{\dagger}(\psi_1)a^{\dagger}(\psi_2)\ldots a^{\dagger}(\psi_n)|0\rangle$, where $a^{\dagger}(\psi_i)$ represents the creation operator for the state $|\psi_i\rangle$
  • The order of the creation operators is important due to the anti-commutation relations
  • Example: $|\uparrow, \downarrow\rangle = a^{\dagger}_{\uparrow}a^{\dagger}_{\downarrow}|0\rangle$ represents a state with two fermions, one with spin up and one with spin down

Expectation values of observables

Expressing observables in terms of creation and annihilation operators

• Observables in second quantization can be expressed in terms of creation and annihilation operators
• Examples of observables include:
  • Number operator: $N = a^{\dagger}a$
  • Hamiltonian: $H = \sum_{ij}h_{ij}a^{\dagger}_ia_j + \frac{1}{2}\sum_{ijkl}V_{ijkl}a^{\dagger}_ia^{\dagger}_ja_la_k$, where $h_{ij}$ and $V_{ijkl}$ are matrix elements
  • Correlation functions: $\langle a^{\dagger}(x)a(y)\rangle$ for bosons or $\langle a^{\dagger}(x)a(y)\rangle$ for fermions

Calculating expectation values

• The expectation value of an observable $A$ in a state $|\psi\rangle$ is given by:
  • $\langle A\rangle = \langle\psi|A|\psi\rangle$, where $\langle\psi|$ is the Hermitian conjugate of $|\psi\rangle$
• To calculate the expectation value:
  1. Express the observable $A$ in terms of creation and annihilation operators
  2. Write the state $|\psi\rangle$ as a linear combination of Fock states or construct it using creation operators acting on the vacuum state
  3. Use the commutation or anti-commutation relations to simplify the expressions
  4. Evaluate the expectation value by applying the operators to the state and taking the inner product
• Example: For a single-particle state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, the expectation value of the number operator is:
  • $\langle N\rangle = \langle\psi|a^{\dagger}a|\psi\rangle = |\beta|^2$, which represents the probability of finding one particle in the state