Second quantization of quantum mechanics
In my exploration of second quantization, I explore the advanced framework that efficiently handles many interacting particles using creation and annihilation operators. This formalism extends the quantum mechanical treatment of particles, allowing for a more streamlined description of states and observables. I focus on the wavefunction operator for fermion systems, showing how it naturally incorporates antisymmetry for multi-particle states. By extending the field operator to multiple particles, I demonstrate the elegant handling of particle exchange, adhering to the Pauli exclusion principle.
In first quantization, the transition from classical mechanics to quantum mechanics involves converting classical Hamiltonian H({\mathbf{r}_i}, {\mathbf{p}_i}) into an operator form.
Second quantization reformulates quantum mechanics to handle many interacting particles more effectively. It uses creation (\mathbf a^\dagger for bosons and \mathbf b^\dagger for fermions) and annihilation (\mathbf a and \mathbf b) operators to manage particle states, obeying commutation or anticommutation rules for bosons and fermions, respectively. The formalism extends to the occupation number representation, where states are described by the number of particles in each quantum state. Field operators \hat{\psi}^\dagger(\mathbf{r}) and \hat{\psi}(\mathbf{r}) are introduced to create and annihilate particles at position \mathbf{r}.
Starting from a single-fermion state, the wavefunction operator for a fermion system can be expressed using the creation (\mathbf{b}^\dagger) and annihilation (\mathbf{b}) operators. For a single fermion state, the wavefunction operator \hat{\psi}(\mathbf{r}) is postulated as:
\hat{\psi}(\mathbf{r}) = \sum_j \phi_j(\mathbf{r}) \mathbf{b}_j
The wavefunction operator for the state with a single fermion in the k^{th} mode is simply the single-particle wavefunction \phi_j(\mathbf{r}) acting on the vacuum state.
For a two-particle system, the wavefunction operator \hat{\psi}(\mathbf{r}_1, \mathbf{r}_2) can be defined in terms of the creation and annihilation operators for two particles. The field operator for a two-particle state is defined as:
\hat{\psi}(\mathbf{r}_1, \mathbf{r}_2) = \frac{1}{\sqrt 2}\sum_{j,k} \phi_j(\mathbf{r}_1)\phi_k(\mathbf{r}_2) \mathbf{b}_k \mathbf{b}_j
Here, \phi_{j,k}(\mathbf{r}_1, \mathbf{r}_2) represents the two-particle wavefunctions (modes), and \mathbf{b}_j and \mathbf{b}_k are the annihilation operators for the j^{th} and k^{th} modes, respectively. The square root factor in front ensures that the solution is normalized.
By extending the field operator to multiple particles, I demonstrate how it correctly handles the permutation for particle exchange. For N particles, the field operator can be extended as follows:
\hat{\psi}(\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N) = \frac{1}{\sqrt N} \sum_{1, 2, \dots, N} \phi_1(\mathbf{r}_1) \phi_2(\mathbf{r}_2) \dots \phi_n(\mathbf{r}_N) \mathbf{b}_{1} \mathbf{b}_2 \dots \mathbf{b}_N
This operator deals correctly with the permutation for particle exchange. This formalism is crucial for understanding the quantum mechanical behavior of systems with multiple particles, providing a natural way to incorporate the antisymmetry of fermions as mandated by the Pauli exclusion principle.
For more insights into this topic, you can find the details here.