Can the Time-Energy Uncertainty Relation Simplify the Schrodinger Equation?

Wannabe Physicist
Messages
17
Reaction score
3
Homework Statement
A particle moves in a 1D symmetric infinite square well of length ##2a##. Inside the well, the potential is ##V(x) = \gamma \delta(x)##. For sufficiently large ##\gamma##, calculate the time required for the particle to tunnel from being in the ground state of the well extending from ##x = -a## to ##x = 0## to the ground state of the well extending from ##x=0## to x=a##.
Relevant Equations
##\Delta t \Delta E \geq \hbar/2##
I am guessing time-energy uncertainty relation is the way to solve this. I solved the Schrodinger equation for both the regions and used to continuity at ##x=-a, 0,a## and got ##\psi(-a<x<0) = A\sin(\kappa(x+a))## and ##\psi(0<x<a) = -A\sin(\kappa(x-a))## where ##\kappa^2 = 2mE/\hbar^2##. Looking at the discontuity in the derivative, I got the transcendental equation
$$\tan(\kappa a) = -\frac{\hbar^2 \kappa}{m\gamma}$$
Setting ##x = \kappa a##,
$$\tan(x) = -\frac{\hbar^2 }{m\gamma a}x$$.

Imitating what one does for finite square well case, I plotted both the curves. One root is at ##x=0##. But that would mean ##\kappa = E = 0##. So that cannot represent a bound state. Next root for ##\gamma a = m/\hbar^2## (which means ##\tan(x) = -x##) is ##x=2.029##. This implies
$$\kappa = \frac{2.029}{a} = 2.029\frac{\hbar^2\gamma}{m}$$
This gives
$$E = \frac{2m\kappa^2}{\hbar^2} = \frac{2m}{\hbar^2}\left(\frac{4.117 \hbar^4\gamma^2}{m^2}\right) = \frac{8.234\hbar^2\gamma^2}{m}$$.

Now ##\Delta t \Delta E \geq \hbar/2##. But $$E = p^2/2m \implies \Delta E = p\Delta p/m= p\hbar/(2m \Delta x) = p\hbar/(2m a) = \sqrt{2mE} \hbar/(2ma) = \sqrt{\frac{2E}{ma^2}}\frac{\hbar}{2}$$

Thus $$\Delta E = \sqrt{\frac{2E}{ma^2}}\frac{\hbar}{2}$$

Finally,
$$\Delta t = \frac{\hbar}{2\Delta E} = \sqrt{\frac{ma^2}{2E}}$$

Substituting ##E = \frac{8.234\hbar^2\gamma^2}{m}## I get
$$\Delta t = 0.4964\frac{ma\gamma}{\hbar}$$

Does all of this make sense? Is there a simpler method and I am over-complicating an extremely simple problem?
 
Physics news on Phys.org
I observe physical dimension of your result is
dim[\frac{ma\gamma}{\hbar}]=ML^2T^{-1}
, not T which we expect. Instead I see
dim[\frac{a\hbar}{\gamma}]=T
Is it helpful for checking your result ?
 
Last edited:
Yes. Thank you.
 
Thread 'Need help understanding this figure on energy levels'
This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
Thread 'Understanding how to "tack on" the time wiggle factor'
The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
Back
Top