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Chapter 5: Q59E (page 191)
Determine the expectation value of the position of a harmonic oscillator in its ground state.
The expectation value of harmonic oscillator in ground state is 0.
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Verify that is a solution of equation .
In Section 5.5, it was shown that the infinite well energies follow simply from the formula for kinetic energy, p2/2m; and a famous standing-wave condition, . The arguments are perfectly valid when the potential energy is 0(inside the well) and L is strictly constant, but they can also be useful in other cases. The length L allowed the wave should be roughly the distance between the classical turning points, where there is no kinetic energy left. Apply these arguments to the oscillator potential energy, .Find the location x of the classical turning point in terms of E; use twice this distance for L; then insert this into the infinite well energy formula, so that appears on both sides. Thus far, the procedure really only deals with kinetic energy. Assume, as is true for a classical oscillator, that there is as much potential energy, on average, as kinetic energy. What do you obtain for the quantized energies?
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 ifis. 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.
The potential energy shared by two atoms in a diatomic molecule, depicted in Figure 17, is often approximated by the fairly simple function where constants a and b depend on the atoms involved. In Section 7, it is said that near its minimum value, it can be approximated by an even simpler function—it should “look like” a parabola. (a) In terms ofa and b, find the minimum potential energy U (x0) and the separation x0 at which it occurs. (b) The parabolic approximation has the same minimum value at x0 and the same first derivative there (i.e., 0). Its second derivative is k , the spring constant of this Hooke’s law potential energy. In terms of a and b, what is the spring constant of U (x)?
Show that the uncertainty in the momentum of a ground state harmonic oscillator is .
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