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Q47E

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Modern Physics
Found in: Page 189
Modern Physics

Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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Short Answer

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 -E0.

(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.

See the step by step solution

Step by Step Solution

Step 1: Given data

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 (-E0).

Step 2: Wave function outside a finite potential well

The wave function of a particle of mass m and energy E outside a finite potential well of height U0 is

.....(II)

Here is the reduced Planck's constant.

 Step 3: Determining the wave function for the delta well potential

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

Step 4: Plotting the wave function

The wave function obtained above is plotted as follows

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

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