2.5 | SUMMARY
Section 2.2 Section 2.2 Schrodinger’s Wave Equation
2.16 Assume that 1(x, t) and 2(x, t) are solutions of the one-dimensional time-dependent Schrodinger’s wave equation. ( a ) Show that 12 is a solution. ( b ) Is 1ⴢ2 a solution of the Schrodinger’s equation in general? Why or why not?
2.17 Consider the wave function (x, t) A cos _ x
2 e jt for 1 x 3.
Determine A so that
1
3
(x, t)2dx 1.
2.18 Consider the wave function (x, t) A(cos nx) e jt for 12 x 12, where n is an integer. Determine A so that
12
12
(x, t)2dx 1.
2.19 The solution to Schrodinger’s wave equation for a particular situation is given by (x) ____2a0 ⴢ e x/a0 . Determine the probability of fi nding the particle between the following limits: ( a ) 0 x a04, ( b ) a04 x a02, and ( c ) 0 x a0. 2.20 An electron is described by a wave function given by (x)
__ 2 _ a cos _ a xfor _ a
2 x _ a
2 . The wave function is zero elsewhere. Calculate the probability of fi nding the electron between (a) 0 x _ a
4 , (b) _ a 4 x _ a
2 , and (c) _ a
2 x _ a 2 . 2.21 Repeat Problem 2.20 if the wave function is given by (x)
__ 2 _ a sin 2_ a x .Section 2.3 Applications of Schrodinger’s Wave Equation
2.22 (a) An electron in free space is described by a plane wave given by (x, t) A e j(kxt) . If k 8 108 m1 and 8 1012 rad/s, determine the (i) phase velocity and wave- length of the plane wave, and the (ii) momentum and kinetic energy (in eV) of the electron. (b) Repeat part (a) for k 1.5 109 m1 and 1.5 1013 rad/s.
2.23 An electron is traveling in the negative x direction with a kinetic energy of 0.025 eV.
(a) Write the equation of a plane wave that describes this particle. (b) What is the wave number, wavelength, and angular frequency of the wave that describes this electron.
2.24 Determine the wave number, wavelength, angular frequency, and period of a wave function that describes an electron traveling in free space at a velocity of (a) v 5 106 cm/s and (b) v 108 cm/s.
2.25 An elecron is bound in a one-dimesional infi nite potential well with a width of 75 Å.
Determine the electron energy levels (in eV) for n 1, 2, 3.
2.26 An electron is bound in a one-dimensional infi nite potential well with a width of 10 Å. (a) Calculate the fi rst three energy levels that the electron may occupy. (b) If the electron drops from the third to the second energy level, what is the wavelength of a photon that might be emitted?
2.27 A particle with a mass of 15 mg is bound in a one-dimensional infi nite potential well that is 1.2 cm wide. (a) If the energy of the particle is 15 mJ, determine the value of n for that state. (b) What is the energy of the (n1) state? (c) Would quantum effects be observable for this particle?
2.28 Calculate the lowest energy level for a neutron in a nucleus, by treating it as if it were in an infi nite potential well of width equal to 10 −14 m. Compsare this with the lowest energy level for an electron in the same infi nite potential well.
2.29 Consider the particle in the infi nite potential well as shown in Figure P2.29. Derive and sketch the wave functions corresponding to the four lowest energy levels. (Do not normalize the wave functions.)
*2.30 Consider a three-dimensional infi nite potential well. The potential function is given by V(x) 0 for 0 x a, 0 y a, 0 z a, and V(x) elsewhere. Start with
*Asterisks next to problems indicate problems that are more diffi cult.
Schrodinger’s wave equation, use the separation of variables technique, and show that the energy is quantized and is given by
E n xn ynz _ 22
2ma2 nx2 ny2 nz2 where n x 1, 2, 3, . . ., n y 1, 2, 3, . . . , n z 1, 2, 3, . . . .
2.31 Consider a free electron bound within a two-dimensional infi nite potential well de- fi ned by V 0 for 0 x 40 Å, 0 y 20 Å, and V elsewhere. ( a ) Determine the expression for the allowed electron energies. ( b ) Describe any similarities and any differences with the results of the one-dimensional infi nite potential well.
2.32 Consider a proton in a one-dimensional infi nite potential well shown in Figure 2.6.
( a ) Derive the expression for the allowed energy states of the proton. ( b ) Calculate the energy difference (in units of eV) between the lowest possible energy and the next higher energy state for ( i ) a 4 Å, and ( ii ) a 0.5 cm.
2.33 For the step potential function shown in Figure P2.33, assume that E V 0 and that particles are incident from the x direction traveling in the x direction. ( a ) Write the wave solutions for each region. ( b ) Derive expressions for the transmission and refl ec- tion coeffi cients.
2.34 Consider an electron with a kinetic energy of 2.8 eV incident on a step potential func- tion of height 3.5 eV. Determine the relative probability of fi nding the electron at a dis- tance (a) 5 Å beyond the barrier, (b) 15 Å beyond the barrier, and (c) 40 Å beyond the barrier compared with the probability of fi nding the incident particle at the barrier edge.
2.35 (a) Calculate the transmission coeffi cient of an electron with a kinetic energy of 0.1 eV impinging on a potential barrier of height 1.0 eV and a width of 4 Å (b) Repeat part (a) for a barrier width of 12 Å. (c) Using the results of part (a), determine the density of electrons per second that impinge the barrier if the tunneling current den- sity is 1.2 mA/cm2.
2.36 ( a ) Estimate the tunneling probability of a particle with an effective mass of 0.067 m 0 (an electron in gallium arsenide), where m 0 is the mass of an electron, tun- neling through a rectangular potential barrier of height V 0 0.8 eV and width 15 Å.
The particle kinetic energy is 0.20 eV. ( b ) Repeat part ( a ) if the effective mass of the particle is 1.08m0 (an electron in silicon).
Figure P2.33 | Potential function for Problem 2.33.
Incident particles V0
x 0 x
V(x) V(x)
x 0 x a
2 x a
2
Figure P2.29 | Potential function for Problem 2.29.
Problems 55
2.37 (a) A proton with a kinetic energy of 1 MeV is incident on a potential barrier of height 12 MeV and width 1014 m. What is the tunneling probability. (b) The width of the potential barrier in part (a) is to be decreased so that the tunneling probability is in- creased by a factor of 10. What is the width of the potential barrier?
*2.38 An electron with energy E is incident on a rectangular potential barrier as shown in Figure 2.9. The potential barrier is of width a and height V0 E. ( a ) Write the form of the wave function in each of the three regions. ( b ) For this geometry, determine what coeffi cient in the wave function solutions is zero. ( c ) Derive the expression for the transmission coeffi cient for the electron (tunneling probability). ( d ) Sketch the wave function for the electron in each region.
*2.39 A potential function is shown in Figure P2.39 with incident particles coming from with a total energy E V2. The constants k are defi ned as
k1
_____ 2mE _ 2 k2 ___________2m
_ 2 (E V1) k3
___________2m _ 2 (E V2) Assume a special case for which k2a 2n, n 1, 2, 3, . . . . Derive the expression, in terms of the constants, k1, k2, and k3, for the transmission coeffi cient. The transmis- sion coeffi cient is defi ned as the ratio of the fl ux of particles in region III to the inci- dent fl ux in region I.
*2.40 Consider the one-dimensional potential function shown in Figure P2.40. Assume the total energy of an electron is E V0. ( a ) Write the wave solutions that apply in each region. ( b ) Write the set of equations that result from applying the boundary conditions. ( c ) Show explicitly why, or why not, the energy levels of the electron are quantized.