Homework Statement
Consider an electron with spin, which should be in a homogenous magnetic field B=B0ez. This situation is described by the Hamiltonian of the shape \hat{H}=g\frac{\mu_B}{\hbar}\textbf{BS}.
Consider now the time dependent state |\psi(t)> of the electron in spin space. The...
I think I got <V>, the first solution I posted was from the indefinite integral.
A^2\frac{e_0^2}{\epsilon_0}\int_0^{\infty}\exp(-2\alpha~r)r~dr
A^2\frac{e_0^2}{\epsilon_0}\cdot\frac{1}{4\alpha^2}=\frac{e_0^2\alpha}{4\pi\epsilon_0}.
Also, A can be found by normalizing the original wave...
I think I got it now. First of all I learned that the 2p energy level is degenerate, thus m should have no influence on it and there is only one calculation to do for 2p.
Then I tried to calculate 1s:
\psi_{1s}=\frac{1}{\sqrt{\pi}}(\frac{Z}{a_0})^{3/2}\exp(-\frac{Zr}{a_0})
As we are...
I also attempted a run at the 2p (for m=0) and it looks like this:
\psi_{2p_0}^2=\frac{1}{32\pi\cdot a_0^5}\cdot r^2Z^5\exp(-\frac{2Z\cdot r}{a_0})\cdot\cos^2(\theta)
<r>=\frac{1}{32\pi\cdot a_0^5}\int_0^r \int_0^{\pi} \int_0^{2\pi} r^2Z^5\exp(-\frac{2Z\cdot...
Homework Statement
Determine for the hydrogen atom states 1s and 2p the expectation value of the radius r and the associated mean square error Δr.
Homework Equations
Wave Functions for 1s and 2p from Demtroeder's Experimental Physics Volume 3 (it says "The normalized complete...
Sorry, I had once seen this notation (T=\frac{\hbar}{2m}\Delta) somewhere. Would it rather be:
<T>=-\int\int\int r^2\cdot\sin(\theta)\cdot\psi^2\cdot\frac{\hbar}{2m}\Delta\psi~dr~d\theta~d\phi ?
I'll try the limits later on, but thanks for the help already :)!
So, I am getting this:
-4\pi\cdot A^2\frac{e_0^2}{4\pi\epsilon_0}\int \exp(-\alpha\cdot r)\cdot r~dr
-A^2\frac{e_0^2}{\epsilon_0}\int \exp(-\alpha\cdot r)\cdot r~dr
and
A^2\frac{e_0^2}{\epsilon_0}\cdot \frac{1-\exp(-\alpha\cdot r)\cdot (\alpha\cdot r+1)}{\alpha^2}
Is this correct? If...
Thank you Vela! Would it maybe be possible for one of you to show me just how to correctly do this calculation, because my problem is that I usually need a good example and explanation on how to calculate complex things. When I look at calculations in books, those are usually good, but when I...
"Optimizing" a Wave Function
Homework Statement
Consider a Hydrogen Atom, an electron in an attractive Coulomb potential of the form V(r)=-\frac{e_0^2}{4\pi\epsilon_0r}, where e0 is the elementary charge. Assume the following wave function for the electron (with α>0):
\psi(r)=Ae^{-\alpha...
Homework Statement
Consider a potential well with infinite high walls, i.e. V(x)=0 for -L/2\leq x \leq +L/2 and V(x)=\infty at any other x.
Consider this problem (the first task was to solve the stationary Schroedinger equation, to get the Energies and Wave Functions, especially for n=1 and...
Ah, thank you! When I take the complex conjugate of ψ for taking the square, then I get
\rho(x,t=0)=\frac{1}{\sqrt{2\pi}d}\exp[-\frac{(x-x_0)^2}{2d^2}].
Looks like a gaussian bell curve to me from the form of the equation. Could that be it?
An addendum, just received an E-Mail from the professor, who said that due to many people having difficulties, he would give the solution for ψ, and he said it looks like this:
\psi(x,0)=\frac{1}{\pi^{1/4}\cdot 2^{1/4}\cdot d_0^{1/2}}\exp[i\cdot k_0(x-x_0)]\exp[-\frac{(x-x_0)^2}{4d_0^2}]...
Homework Statement
Consider the time-dependent one-dimensional Schroedinger Equation for the free particle, i.e. let the Potential be V(x)=0. Consider a wave packet, i.e.
\psi(x,t)=\int_{-\infty}^{\infty}=A(k)\exp[i(kx-\omega(k)t]dk.
Consider especially the Amplitude distribution...
Homework Statement
Charge Q1 is fixed to the ceiling by means of a spring with the spring constant D and is moved from its rest position (which is at y=h) by the two charges Q2. The gravitational force may be neglected.
a) Determine the resulting force on Q1 in dependency of y.
b) The...