Consider a cubic 3D infinite well.
(a) How many different wave functions have the same energy as the one for which ?
(b) Into how many different energy levels would this level split if the length of one side were increased by ?
(c) Make a scale diagram, similar to Figure 3, illustrating the energy splitting of the previously degenerate wave functions.
(d) Is there any degeneracy left? If so, how might it be “destroyed”?
(a) There are two other wave functions having the same energy as the one for which .
(b) This energy level would split into two new energy levels if the length of one side is increased by .
(c) The scale diagram is as follows
(d) There will still be two wave functions having the same energy. This can be removed by changing any of the side lengths by a small amount.
There is a 3D infinite cubic well.
The energy of a particle of mass m in a 3D infinite well of sides , and is
Here, is the reduced Planck's constant and , and are the quantum numbers
For a cubic well, the energy in equation (I) reduces to
Thus the energy when any one of is 5 and the rest two are 1 are the same. Hence the wave functions corresponding to , and have the same energies. There are three wave functions having the same energy.
Let the length along X axis, that is be increased by . The new length is then . The energy in equation (II) becomes
Thus the wave function corresponding to will now have lower energy than the wave functions corresponding to and . The previous energy level will thus split into two separate energy levels.
The plot of the initial and final energy levels are as follows
There are still two wave functions corresponding to and that are in the same energy level. Thus a degeneracy of 2 is left. This can be removed if one of the lengths along x or y axis is slightly changed making the energy equation equal to equation (I).
Mathematically equation (7-22) is the same differential equation as we had for a particle in a box-the function and its second derivative are proportional. But for m1 = 0 is a constant and is allowed, whereas such a constant wave function is not allowed for a particle in a box. What physics accounts for this difference?
Classically, an orbiting charged particle radiates electromagnetic energy, and for an electron in atomic dimensions, it would lead to collapse in considerably less than the wink of an eye.
(a) By equating the centripetal and Coulomb forces, show that for a classical charge -e of mass m held in a circular orbit by its attraction to a fixed charge +e, the following relationship holds
(b) Electromagnetism tells us that a charge whose acceleration is a radiates power . Show that this can also be expressed in terms of the orbit radius as . Then calculate the energy lost per orbit in terms of r by multiplying the power by the period and using the formula from part (a) to eliminate .
(c) In such a classical orbit, the total mechanical energy is half the potential energy, or . Calculate the change in energy per change in r : . From this and the energy lost per obit from part (b), determine the change in per orbit and evaluate it for a typical orbit radius of . Would the electron's radius change much in a single orbit?
(d) Argue that dividing by P and multiplying by dr gives the time required to change r by dr . Then, sum these times for all radii from to a final radius of 0. Evaluate your result for . (One limitation of this estimate is that the electron would eventually be moving relativistically).
A comet of mass describes a very elliptical orbit about a star of mass , with its minimum orbit radius, known as perihelion, being role="math" localid="1660116418480" and its maximum, or aphelion, times as far. When at these minimum and maximum
radii, its radius is, of course, not changing, so its radial kinetic energy is , and its kinetic energy is entirely rotational. From classical mechanics, rotational energy is given by , where is the moment of inertia, which for a “point comet” is simply .
(a) The comet’s speed at perihelion is . Calculate its angular momentum.
(b) Verify that the sum of the gravitational potential energy and rotational energy are equal at perihelion and aphelion. (Remember: Angular momentum is conserved.)
(c) Calculate the sum of the gravitational potential energy and rotational energy when the orbit radius is times perihelion. How do you reconcile your answer with energy conservation?
(d) If the comet had the same total energy but described a circular orbit, at what radius would it orbit, and how would its angular momentum compare with the value of part (a)?
(e) Relate your observations to the division of kinetic energy in hydrogen electron orbits of the same but different .
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