Schrödinger

[logistic_guy]

\(\displaystyle \bold{(a)}\) Show that \(\displaystyle \Psi(x,t) = Ae^{i(kx-\omega t)}\) is a solution to the time-dependent Schrödinger equation for a free particle \(\displaystyle [U(x) = U_0 = \ \text{constant}]\) but that \(\displaystyle \Psi(x,t) = A\cos(kx - \omega t)\) and \(\displaystyle \Psi(x,t) = A\sin(kx - \omega t)\) are not.

\(\displaystyle \bold{(b)}\) Show that the valid solution of part (a) satisfies conservation of energy if the de Broglie relations hold, \(\displaystyle \lambda = h/p, \ \omega = E/\hbar\). That is, show that direct substitution in the time-dependent Schrödinger equation gives \(\displaystyle \hbar\omega = \frac{\hbar^2 k^2}{2m} + U_0\).

 

[logistic_guy]

If you don't understand Quantum Mechanics, you are missing the joy of your half-life

Let us start by writing down the one-dimensional time-dependent Schrödinger equation.

\(\displaystyle -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + U(x)\Psi = i\hbar\frac{\partial \Psi}{\partial t}\)

It looks cute, isn't it?I told you that you're missing a lot.

Don't panic baby if you don't understand this equation! Just imagine yourself looking at any ordinary energy equation.

We will investigate more on this in later posts. Keep coming back to this thread if you wanna enjoy some of your time (and life).