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Chapter 2: Q19P (page 66)
Question: Find the probability current, (Problem 1.14) for the free particle wave function Equation . Which direction does the probability flow?
The probability current is
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Imagine a bead of mass m that slides frictionlessly around a circular wire ring of circumference L. (This is just like a free particle, except that find the stationary states (with appropriate normalization) and the corresponding allowed energies. Note that there are two independent solutions for each energy En-corresponding to clockwise and counter-clockwise circulation; call them and How do you account for this degeneracy, in view of the theorem in Problem 2.45 (why does the theorem fail, in this case)?
If two (or more) distinct44solutions to the (time-independent) Schr枚dinger equation have the same energy E . These states are said to be degenerate. For example, the free particle states are doubly degenerate-one solution representing motion to the right. And the other motion to the left. But we have never encountered normalizable degenerate solutions, and this is no accident. Prove the following theorem: In one dimension45 there are no degenerate bound states. Hint: Suppose there are two solutions, and with the same energy E. Multiply the Schr枚dinger equation for by and the Schr枚dinger equation for by and subtract, to show that is a constant. Use the fact that for normalizable solutions to demonstrate that this constant is in fact zero.Conclude that s a multiple of and hence that the two solutions are not distinct.
Prove the following three theorem;
a) For normalizable solutions the separation constant E must be real as and show that if equation 1.20 is to hold for all must be zero.
b) The time - independent wave function localid="1658117146660" can always be taken to be real, This doesn鈥檛 mean that every solution to the time-independent Schrodinger equation is real; what it says is that if you鈥檝e got one that is not, it can always be expressed as a linear combination of solutions that are . So, you might as well stick to 鈥檚 that are real
c) If is an even function then can always be taken to be either even or odd
a) Show that the wave function of a particle in the infinite square well returns to its original form after a quantum revival time T = 4ma2/蟺~. That is: 唯 (x, T) = 唯 (x, 0) for any state (not just a stationary state).
(b) What is the classical revival time, for a particle of energy E bouncing back and forth between the walls?
(c) For what energy are the two revival times equal?
Analyze the odd bound state wave functions for the finite square well. Derive the transcendental equation for the allowed energies and solve it graphically. Examine the two limiting cases. Is there always an odd bound state?
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