Equation gives infinite well energies. Because equation cannot be solved in closed form, there is no similar compact formula for finite well energies. Still many conclusions can be drawn without one. Argue on largely qualitative grounds that if the walls of a finite well are moved close together but not changed in height, then the well must progressively hold fewer bound states. (Make a clear connection between the width of the well and the height of the walls).
When the walls of our infinite well are drawn closer together, the wavelengths of standing waves shorten, and the particle is no longer imprisoned in that condition.
The wavelengths of the standing waves get shorter when the walls of our infinite well are pulled closer together, suggesting greater momentum and, in turn, kinetic energy. The kinetic energies of some states will exceed the height of the potential energy walls as the walls become closer together, and the particle will no longer be trapped in that state.
When the walls of our infinite well are brought closer together, the wavelengths of the standing waves shorten, implying more momentum and, hence, kinetic energy. As the potential energy barriers move closer together, the kinetic energies of some states will exceed the height of the potential energy walls, and the particle will no longer be imprisoned in that state.
As the walls are brought together the particle will no longer be imprisoned in that state.
If a particle in a stationary state is bound, the expectation value of its momentum must be 0.
(a). In words, why?
(b) Prove it.
Starting from the general expression(5-31) with in the place of , integrate by parts, then argue that the result is identically 0. Be careful that your argument is somehow based on the particle being bound: a free particle certainly may have a non zero momentum. (Note: Without loss of generality, may be chosen to be real.)
Show that that is, verify that unless the wave function is an Eigen function of the momentum operator, there will be a nonzero uncertainty in the momentumstarts with showing that the quantity
Is . Then using the differential operator form ofand integration by parts, show that it is also,
Together these show that if is. 0. then the preceding quantity must be 0. However, the Integral of the complex square of a function(the quantity in the brackets) can only be 0 if the function is identically 0, so the assertion is proved.
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