Friday 17 January 2020

quantum mechanics - Am I missing a trick to solving a 3D potential well problem?



I was playing around with a 3-D potential $V$ such that $V_{(r)} = 0$ for $r0$ otherwise. By using the Schrödinger Equation, I showed that: $$\frac{-\hbar}{2m}\frac{1}{r^2}\frac{d}{dr}\bigl( r^2\frac{d}{dr}\bigr)\psi = E\psi$$


I then used the substitution $\psi_{(r)}=f_{(r)}/r$ and $k=\sqrt{2mE}/\hbar$ to get: $$\frac{1}{r}\frac{d^2f_{(r)}}{dr^2}=-\frac{k^2}{r}f_{(r)} \tag{I}$$


which describes the wavefunction $\psi_{(r)}=f_{(r)}/r$ inside the sphere. Hence, the differential equation has the domain $0\leq r

If I do the same thing to $(I)$, I obtain the equation for simple harmonic motion, but substituting the solution back into $(I)$ as a sanity check gives a division by zero when evalutating for $r=0$. After that, I tried a number of substitutions to make $(I)$ have a more recognisable form - to no avail. Then I had the idea of multiplying my trial solution by some other function of $r$ so that upon substitution into $(I)$, the evaluation of $r=0$ doesn't give an infinity... but I don't know quite how to do that...


Long story short..... my question is: what trick do I need to get a meaningful solution to $(I)$?



Answer



I) The substitution $f=r\psi$ is the standard substitution to get a radial 3D problem to resemble a 1D problem, see e.g. Ref. 1.


II) From the perspective of the normalization of the wavefunction $\psi(r)$, a $1/r$ singularity of $\psi(r)$ at $r=0$ is fine because $|\psi(r)|^2$ is suppressed by a Jacobian factor $r^2$ coming from the measure in 3D spherical coordinates.


III) However in order to keep the kinetic energy $K=\frac{\hbar^2}{2m} \int d^3x~ |\nabla \psi|^2$ finite, a $1/r$ singularity of $\psi(r)$ at $r=0$ is unacceptable, i.e., we must discard the cosine solution and only keep the sine solution. This corresponds to imposing that the wavefunction $\psi(r)$ should be bounded.


References:




  1. D. Griffiths, Intro to QM, Section 4.1.3.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...