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Chapter 39: Q57P (page 1217)
An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference between its quantum levels n and n+2 is .
It is proved that .
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The wave functions for the three states with the dot plots shown in Fig. 39-23, which have n = 2 , l = 1 , and 0, and , are
in which the subscripts on give the values of the quantum numbers n , l , and the angles and are defined in Fig. 39-22. Note that the first wave function is real but the others, which involve the imaginary number i, are complex. Find the radial probability density P(r) for (a) and (b) (same as for ). (c) Show that each P(r) is consistent with the corresponding dot plot in Fig. 39-23. (d) Add the radial probability densities for , , and and then show that the sum is spherically symmetric, depending only on r.
A neutron with a kinetic energy of 6.0 eV collides with a stationary hydrogen atom in its ground state. Explain why the collision must be elastic—that is, why kinetic energy must be conserved. (Hint: Show that the hydrogen atom cannot be excited as a result of the collision.)
An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energy difference equal to the energy of the n = 5 level? (c) Show that no pair of adjacent levels has an energy difference equal to the energy of the n = 6 level.
A hydrogen atom in a state having a binding energy (the energy required to remove an electron) of 0.85 eV makes a transition to a state with an excitation energy (the difference between the energy of the state and that of the ground state) of . (a) What is the energy of the photon emitted as a result of the transition? What are the (b) higher quantum number and (c) lower quantum number of the transition producing this emission?
As Fig. 39-8 suggests, the probability density for the region X>L in the finite potential well of Fig. 39-7 drops off exponentially according to , where C is a constant. (a) Show that the wave function that may be found from this equation is a solution of Schrödinger’s equation in its one-dimensional form. (b) Find an expression for k for this to be true.
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