Proving Eigenstate Energy: Multiply ψ(x) by Hamiltonian Operator for e-ikx

[TheNE]

show that e^-ikx is an eigenstate energy.

Do I start by multiplying the hamiltonian operator by ψ(x)?

So far I have ψ(x)(1/2m)(-i[STRIKE]h[/STRIKE]d/dx)²=e^-ikx

 

[genericusrnme]

The Hamiltonian usually goes on the other side of $\psi$
Do you know schrodingers equation?

 

[TheNE]

-[STRIKE]h[/STRIKE]²/2m d²/dx² e^ikx-iωt right?

To prove the eigenstate, I could just have Hψ(x)=hψ(x) and break H down into it's smaller parts, right? I thought I took good enough notes on eigenvalues... Tyvm!

 

[TheNE]

Where H is the Hamiltonian operator

 

[genericusrnme]

There's a right hand side to schrodingers equation too :P

 

[vanhees71]

It has not so much to do with the Schrödinger equation, but with the eigenvalue problem for the Hamiltonian $\hat{H}$. In position space the Hamilton operator for a free particle is given by the differential operator
$$\hat{H}=-\frac{\hbar^2}{2m} \Delta.$$
It's indeed easy to schow that the plane wave
$$u_{\vec{k}}(\vec{x})=N \exp(\mathrm{i} \vec{k} \cdot \vec{x})$$
is a generalized eigenfunction of $\hat{H}$ (with $N$ an arbitrary constant). Just take the derivatives and check that it fulfills the eigenvalue equation
$$\hat{H} u_{\vec{k}}(\vec{x})=E_{\vec{k}} u_{\vec{k}}(\vec{x}).$$
You'll easily find the energy eigenvalue.

Also think about, whether this function can ever represent a state of the particle in the sense of quantum theory. To help a bit: The answer is a clear no!

Sometimes the eigenvalue equation for the Hamilton operator is called "the time-independent Schrödinger equation". Indeed, the relation with the Schrödinger equation,
$$\mathrm{i} \hbar \partial_t \psi(t,\vec{x})=\hat{H} \psi(t,\vec{x}),$$
is that the function
$\psi_{\vec{k}}(t,\vec{x})=u_{\vec{k}}(\vec{x}) \exp \left (-\mathrm{i} \frac{t E_{\vec{k}}}{\hbar} \right)$
is a solution. The eigenfunctions of the Hamilton operator represent the "stationary solutions", because it is constant in time up to the phase factor $\exp(-\mathrm{i} t E_{\vec{k}}/\hbar)$.

 

[TheNE]

Thanks guys, I got help from my classmates and they walked me through the solution. I wish quantum mechanics (or what I've been exposed to it thus far in my Modern Physics course) was less "symbolic" and easier to apply. As an engineering student, I always seek ways to apply my knowledge, and this really messes with me haha.