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Results tagged with hamiltonian
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user 7270
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
6
votes
1
answer
780
views
Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
The electron has Hamiltonian
$H=\frac{1}{2m}((P_1-\frac{1}{2}eBX_2)^2+(P_2+\frac{1}{2}eBX_1)^2)$
How can I show that this is analogous to the one dimensional harmonic oscillator and then use this fact … I have attempted to expand out the Hamiltonian and found:
$(\frac{P^2_1}{2m}+ \frac{1}{2} m (\frac{eB}{2m}))^2X^2_1+(\frac{P^2_2}{2m}+\frac{1}{2}m(\frac{eB}{2m})^2)X^2_2+\frac{eB}{2m}(X_1P_2-P_1X_2)$ …
- 775
9
votes
3
answers
28k
views
How to express a Hamiltonian operator as a matrix?
Suppose we have Hamiltonian on $\mathbb{C}^2$
$$H=\hbar(W+\sqrt2(A^{\dagger}+A)).$$
We also know $AA^{\dagger}=A^{\dagger}A-1$ and $A^2=0$, letting $W=A^{\dagger}A$. …
- 775
2
votes
1
answer
3k
views
Solving time dependent Schrodinger equation in matrix form
If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector
$$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$
With Hamiltonian $H$ given by
$$H=\hbar\omega
\begin{pmatrix …
- 775