7
votes
4answers
227 views

Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?

Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$ where ...
1
vote
2answers
53 views

What is the energy operator and from where do we get it?

I am trying to learn Quantum mechanics from MIT OCW Videos about quantum mechanics. I have reached the 5th lecture. Please help me in understanding this: In the middle (At 32:08), the professor wrote ...
2
votes
1answer
45 views

Time evolution of states - Is total energy constant or not?

Suppose the state of the particle is given as follows: $$ |\psi_{(t)}\rangle = \frac{1}{\sqrt2} \left( e^{-\frac{i\omega t}{2}} |0\rangle + e^{-\frac{3i\omega t}{2}} |1\rangle \right) $$ Where the ...
10
votes
2answers
227 views

The formal solution of the Schrodinger equation

Let's have Schrodinger equation (or some equation in Schrodinger form) $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{H} \Psi . $$ One likes to write that it has formal solution $$ \tag 2 \Psi (t) ~=~ ...
2
votes
1answer
145 views

Time Dependent Position Operator

How does one find the time dependent position expectation value for a wave function? I thought we could simple take the time dependent wave and apply the position operator like normal, but this gave ...
0
votes
1answer
100 views

How to solve Schrödinger's equation for this potential algebraically?

I want to solve Schrödinger's equation with the potential $$V(x)=\frac{1}{2}mx^2+\lambda x$$ algebraically? Is there any way to construct ladder operators that are similar to the one for the harmonic ...
3
votes
2answers
110 views

Observable Operator on a Superposition?

I'm probably missing something obvious and basic here but I can't make sense of certain usages of Observables as present in basic treatments of Quantum Mechanics that i've come across. $$ ...
1
vote
2answers
184 views

Representation of Hamiltonian in terms of “creation” and “destruction” operators

Let's have Schrodinger equation or Dirac equation in Schrodinger form: $$ i \partial_{0}\Psi = \hat {H}\Psi . $$ Sometimes we can introduce some operators $\hat {A}, \hat {B}$ (the second is not ...
0
votes
1answer
1k views

Expectation value of position in infinite square well

I'm looking for some help to a question. I'm working in the infinite square well, and I have the wavefunction: $$\psi(x,t=0)=A\left( i\sqrt{2}\phi_{1}+\sqrt{3}\phi_{2} \right).$$ For every time t, ...
14
votes
1answer
547 views

Discreteness of set of energy eigenvalues

Given some potential $V$, we have the eigenvalue problem $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$ with the boundary condition $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$ If we ...
1
vote
1answer
91 views

Question about the linearity of wave functions

For piece-wise constant potential, the potential energy is constant so the time dependent wave function can take the form $\psi(x,t)=C_1e^{i(kx- \omega t)}+C_2e^{i(-kx-\omega t)}$ where ...
1
vote
1answer
605 views

The Hermiticity of the Laplacian (and other operators)

Is the Laplacian operator, $\nabla^{2}$, a Hermitian operator? Alternatively: is the matrix representation of the Laplacian Hermitian? i.e. $$\langle \nabla^{2} x | y \rangle = \langle x | ...
4
votes
1answer
409 views

Can we solve the particle in an infinite well in QM using creation and annihilation operators?

The particle in an infinite potential well in QM is usually solved by easily solving Schrodinger differential equation. On the other hand particle in the harmonic oscillator oscillator potential can ...
2
votes
1answer
187 views

Degeneracy and the Hamiltonian

How many linearly independent eigenfunctions can be associated with one degenerate eigenvalue of the Hamiltonian operator? (Is there a limit since it contains a 2nd order differential operator?) ...