1. The problem statement, all variables and given/known data
Find the eigenfunctions and the eigenvalues of the following Hamiltonian
[itex]\hat{H} = \frac{1}{2m} \left ( \frac{ \hbar}{i} \vec{\nabla}-\frac{q}{c}(0, B_z x,0) \right ) ^2 = \frac{1}{2m} \left ( - \hbar^2 \vec{\nabla}^2-\frac{\hbar q}{ic}(\vec{\nabla}·(0, B_z\hat{x},0) + (0,B_z x, 0)·\vec{\nabla}) +\left(\frac{q}{c}\right)^2(0, B_z x,0)^2\right ) ^2 = [/itex]
Assume the wavefunction is of the form [itex]\Psi(x) = e ^{i(k_y y + k_z z)}\chi(x)[/itex]
2. Relevant equations
The time dependent Schrodinger Equation:
[itex]\hat{H} \psi(x) = E \psi (x)[/itex]
3. The attempt at a solution
Computing the scalar products we obtain:
[itex]\hat{H} =\frac{1}{2m} \left ( - \hbar^2 \vec{\nabla}^2-\frac{q}{c}B_z x \frac{\hbar \partial}{i \partial y} + \left(\frac{q}{c}B_z x\right)^2 \right ) ^2 [/itex]
And we can rewrite [itex]\frac{\hbar \partial}{i \partial y}[/itex] as [itex]\hat{p} _y[/itex], so:
[itex]\hat{H} =\frac{1}{2m} \left ( - \hbar^2 \vec{\nabla}^2-\frac{q}{c}B_z x \hat{p}_y + \left(\frac{q}{c}B_z x\right)^2 \right ) ^2 [/itex]
Next I state the time-dependent Shcrodinger Equation:
[itex]\frac{-\hbar ^2}{2m}\vec{\nabla}^2\psi-\frac{q}{2mc}B_z\hat{x}\hat{p}_y \psi+\frac{1}{2m}\left (\frac{q}{c} B_z \hat{x}\right)^2 \psi = E \psi[/itex]
The x that appears along in the Bz term: is it the position operator or an scalar operator with the value x? If its the position operator how does it interact with the wavefunction?
Find the eigenfunctions and the eigenvalues of the following Hamiltonian
[itex]\hat{H} = \frac{1}{2m} \left ( \frac{ \hbar}{i} \vec{\nabla}-\frac{q}{c}(0, B_z x,0) \right ) ^2 = \frac{1}{2m} \left ( - \hbar^2 \vec{\nabla}^2-\frac{\hbar q}{ic}(\vec{\nabla}·(0, B_z\hat{x},0) + (0,B_z x, 0)·\vec{\nabla}) +\left(\frac{q}{c}\right)^2(0, B_z x,0)^2\right ) ^2 = [/itex]
Assume the wavefunction is of the form [itex]\Psi(x) = e ^{i(k_y y + k_z z)}\chi(x)[/itex]
2. Relevant equations
The time dependent Schrodinger Equation:
[itex]\hat{H} \psi(x) = E \psi (x)[/itex]
3. The attempt at a solution
Computing the scalar products we obtain:
[itex]\hat{H} =\frac{1}{2m} \left ( - \hbar^2 \vec{\nabla}^2-\frac{q}{c}B_z x \frac{\hbar \partial}{i \partial y} + \left(\frac{q}{c}B_z x\right)^2 \right ) ^2 [/itex]
And we can rewrite [itex]\frac{\hbar \partial}{i \partial y}[/itex] as [itex]\hat{p} _y[/itex], so:
[itex]\hat{H} =\frac{1}{2m} \left ( - \hbar^2 \vec{\nabla}^2-\frac{q}{c}B_z x \hat{p}_y + \left(\frac{q}{c}B_z x\right)^2 \right ) ^2 [/itex]
Next I state the time-dependent Shcrodinger Equation:
[itex]\frac{-\hbar ^2}{2m}\vec{\nabla}^2\psi-\frac{q}{2mc}B_z\hat{x}\hat{p}_y \psi+\frac{1}{2m}\left (\frac{q}{c} B_z \hat{x}\right)^2 \psi = E \psi[/itex]
The x that appears along in the Bz term: is it the position operator or an scalar operator with the value x? If its the position operator how does it interact with the wavefunction?
0 commentaires:
Enregistrer un commentaire