Role of single-particle fermion operators in multi-particle systems
Following my previous post on second quantization here, this blog post explores the extension of single-particle fermion operators to multi-particle systems within the occupation number formalism.
Extending Single-Particle Operators to Multi-Particle Systems
In quantum mechanics, particularly when dealing with systems composed of multiple particles, it’s essential to extend the analysis from single to multiple particle states. Here, I focus on how to generalize single-particle operators using the occupation number formalism. For an operator \mathbf{\hat G} acting on a single-particle state, the transition to multiple particles involves summing the contributions for each state:
\mathbf{\hat G}_{\mathbf r} = \sum_{a=1}^N \mathbf{G}{\mathbf r_a}
This forms the basis for understanding how operators apply in a system of many fermions.
Mathematical formalism
To illustrate this, consider an operator \mathbf{\hat G} that is defined as:
\mathbf{ \hat G} = \int \hat \psi^\dag \mathbf{G}_{\mathbf r} \hat \psi \mathrm d^3 \mathbf r_1 \dots \mathrm d^3 \mathbf r_N
By substituting the wavefunction operator, I derive the following integral:
\begin{aligned} \mathbf{\hat G} & = \frac{1}{N} \sum_{a=1}^N \sum_{j,k} \mathbf{b}_k^\dag \mathbf{b}_j \int \bar \phi_k(\mathbf{r}) \mathbf{\hat G}_{\mathbf r_a} \phi_j(\mathbf{r}) \mathrm{d}^3\mathbf{r}_1 \dots \mathrm{d}^3\mathbf{r}_N \\ & = \sum_{j,k} \mathbf{b}_k^\dag \mathbf{b}_j G_{ij} \end{aligned}
Analysis of the hamiltonian
The Hamiltonian for a single fermion can be written as:
\mathbf H_{\mathbf r} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf r)
Using the defined multi-particle operator, the Hamiltonian becomes:
\mathbf{ \hat{H}} = \sum_{j} \mathbf{b}_j^\dag \mathbf{b}_j E_j
Here, E_j represents the energy eigenvalues associated with each state. This formulation shows that the Hamiltonian is diagonal in this basis, making it straightforward to calculate the energy for any configuration of particles.
Expectation values of energy
In systems where the states are superposed, calculating the expectation value of the energy becomes important. For a superposition of states:
| \psi \rangle = \sum \alpha_i | \phi_i \rangle
The expectation value is given by:
\langle E \rangle = \sum_i |\alpha_i|^2 E_i
This result aligns with calculations from first quantization, affirming the consistency and power of the second quantization framework.
Conclusion
By extending single-particle operators to systems with multiple fermions, I’ve demonstrated the robustness of the occupation number formalism in handling complex quantum mechanical systems. This approach is not only systematic but also simplifies many calculations, providing a clear pathway for analyzing many-body quantum systems.
For more insights into this topic, you can find the details here.