Suggested languages for you:

Americas

Europe

Q47E

Expert-verifiedFound in: Page 189

Book edition
2nd Edition

Author(s)
Randy Harris

Pages
633 pages

ISBN
9780805303087

**Consider the delta well potential energy: **

** **** **

**Although not completely realistic, this potential energy is often a convenient approximation to a very strong, very narrow attractive potential energy well. It has only one allowed bound-state wave function, and because the top of the well is defined as U = 0, the corresponding bound-state energy is negative. Call its value -E_{0}.**

**(a) Applying the usual arguments and required continuity conditions (need it be smooth?), show that the wave function is given by**

** **** **

**(b) Sketch and U(x) on the same diagram. Does this wave function exhibit the expected behavior in the classically forbidden region?**

(a) It is verified that the wave function for the delta well potential is given by

(b) The plot of the wave function is given below. It exhibits expected behavior in the classically forbidden region.

There is a delta well potential of the form

....................................(I)

The top of the well is defined as U = 0 and the corresponding bound-state energy is negative (-E_{0}).

The wave function of a particle of mass m and energy E outside a finite potential well of height U_{0} is

.....(II)

Here is the reduced Planck's constant.

For the delta potential, = 0 and E = and the wave function in equation (II) reduces to

The function has to be continuous at x = 0 and thus

A = B

The function thus becomes

Normalize this to get

Let

Thus

The final wave function is

The wave function obtained above is plotted as follows

The wave function exponentially falls off in the classically forbidden region as expected.

94% of StudySmarter users get better grades.

Sign up for free