### Corrections to the Matrix Element Evaluation (Can be rendered in markdown): The original expression for evaluating matrix elements is correct in its general form but could benefit from a few mathematical clarifications: 1. **Summation Notation:** If \( H_{nm}(\mathbf{R}) \) includes summation over indices for completeness in the Hilbert space (e.g., a crystal Hamiltonian in a tight-binding model), it should be explicitly indicated. Otherwise, we assume no summation over indices. 2. **Dirac Notation and Completeness:** Ensure that the functions \( \varphi_{n \boldsymbol{R}}(\mathbf{x}) \) are complete and orthonormal if this matrix element definition is to comprehensively represent the system’s Hamiltonian. 3. **Potential \( V(\mathbf{x}) \) Periodicity:** If \( V(\mathbf{x}) \) is periodic, consider shifting it according to \( \mathbf{R} \). This can be achieved by rewriting the expression as: \[ = \int d \mathbf{x} \, \varphi_{s \mathbf{R}}^*(\mathbf{x})\left(-\frac{\hbar^2}{2 m} abla^2 + V(\mathbf{x} - \mathbf{R}) ight) \varphi_{s \mathbf{0}}(\mathbf{x}). \] ### Corrected Expression: With these clarifications, the adjusted expression is: \[ H_{n m}(\mathbf{R}) = \langle \varphi_{n \boldsymbol{R}} | H | \varphi_{m \mathbf{0}} angle = \int d \mathbf{x} \, \varphi_{s \mathbf{R}}^*(\mathbf{x}) \left(-\frac{\hbar^2}{2m} abla^2 + V(\mathbf{x} - \mathbf{R}) ight) \varphi_{s \mathbf{0}}(\mathbf{x}). \] This form maintains the periodicity and translational symmetry, assuming \( \mathbf{R} \) represents a lattice vector.
3:33, there is a small mistake, where wave eq. φ_sR should be φ_s0. And thank you for the valuable video!
### Corrections to the Matrix Element Evaluation (Can be rendered in markdown):
The original expression for evaluating matrix elements is correct in its general form but could benefit from a few mathematical clarifications:
1. **Summation Notation:** If \( H_{nm}(\mathbf{R}) \) includes summation over indices for completeness in the Hilbert space (e.g., a crystal Hamiltonian in a tight-binding model), it should be explicitly indicated. Otherwise, we assume no summation over indices.
2. **Dirac Notation and Completeness:** Ensure that the functions \( \varphi_{n \boldsymbol{R}}(\mathbf{x}) \) are complete and orthonormal if this matrix element definition is to comprehensively represent the system’s Hamiltonian.
3. **Potential \( V(\mathbf{x}) \) Periodicity:** If \( V(\mathbf{x}) \) is periodic, consider shifting it according to \( \mathbf{R} \). This can be achieved by rewriting the expression as:
\[
= \int d \mathbf{x} \, \varphi_{s \mathbf{R}}^*(\mathbf{x})\left(-\frac{\hbar^2}{2 m}
abla^2 + V(\mathbf{x} - \mathbf{R})
ight) \varphi_{s \mathbf{0}}(\mathbf{x}).
\]
### Corrected Expression:
With these clarifications, the adjusted expression is:
\[
H_{n m}(\mathbf{R}) = \langle \varphi_{n \boldsymbol{R}} | H | \varphi_{m \mathbf{0}}
angle = \int d \mathbf{x} \, \varphi_{s \mathbf{R}}^*(\mathbf{x}) \left(-\frac{\hbar^2}{2m}
abla^2 + V(\mathbf{x} - \mathbf{R})
ight) \varphi_{s \mathbf{0}}(\mathbf{x}).
\]
This form maintains the periodicity and translational symmetry, assuming \( \mathbf{R} \) represents a lattice vector.
Thanks for the suggestions!