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Chapter 5: Q34E (page 188)
A 50 electron is trapped between electrostatic walls 200eV high. How far does its wave function extend beyond the walls?
Electron wave extent beyond the walls .
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Outline a procedure for predicting how the quantum-mechanically allowed energies for a harmonic oscillator should depend on a quantum number. In essence, allowed kinetic energies are the particle-in-a box energies, except the length Lis replaced by the distance between classical tuning points. Expressed in terms of E. Apply this procedure to a potential energy of the form U(x) = - b/x where b is a constant. Assume that at the origin there is an infinitely high wall, making it one turning point, and determine the other numing point in terms of E. For the average potential energy, use its value at half way between the tuning points. Again in terms of E. Find and expression for the allowed energies in terms of m, b, and n. (Although three dimensional, the hydrogen atom potential energy is of this form. and the allowed energy levels depend on a quantum number exactly as this simple model predicts.)
In the harmonic oscillators eave functions of figure there is variation in wavelength from the middle of the extremes of the classically allowed region, most noticeable in the higher-n functions. Why does it vary as it does?
The deeper the finite well, the more state it holds. In fact, a new state, the, is added when the well鈥檚 depthreaches. (a) Argue that this should be the case based only on, the shape of the wave inside, and the degree of penetration of the classically forbidden region expected for a state whose energy E is only negligibly below. (b) How many states would be found up to this same 鈥渉eight鈥� in an infinite well.
Determine the expectation value of the momentum of the particle. Explain.
Determine the expectation value of the position of a harmonic oscillator in its ground state.
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