41 1 Model Model the electron as a particle in a rigid one dimensional box of length L Solve Absorption occurs from the ground state n 1 It s reasonable to assume that the transition is from n 1 to n 2 The energy levels of an electron in a rigid box are E n 2 n 2 h 2 8 mL The absorbed photons must have just the right energy so 3 h 2 mL 8 hc E 1 hf E E E elec ph 2 2 L 3 h 8 mc 3 6 63 10 8 9 11 10 34 31 7 J s 6 00 10 m kg 3 0 10 m s 8 7 39 10 10 m 0 739 nm 41 2 Model Model the electron as a particle in a rigid one dimensional box of length L Solve b The energy levels of an electron in a rigid box are a The wavelength 1484 nm is in the infrared range The emitted photons must have just the right energy so E n 2 n 2 h 2 8 mL E ph hf E elec E 3 E 2 hc 2 h 5 2 8 mL L h 5 mc 8 5 6 63 10 8 9 11 10 34 31 9 J s 1484 10 m kg 3 0 10 m s 8 1 5 10 m 1 5 nm 9 41 3 Model Model the electron as a particle in a rigid one dimensional box of length L Solve The energy levels for a particle in a rigid box are nE 2 n 2 2 cid 61 22 L The wave function shown in Figure Ex41 3 corresponds to n 3 This is also shown in Figure 41 7 Thus L cid 61 3 mE 2 3 3 h mE 2 2 3 34 J s 3 6 63 10 kg 6 0 eV 1 6 10 31 19 J eV 2 2 9 11 10 0 75 nm 41 4 Model Model the electron as a particle in a rigid one dimensional box of length L Solve From Equation 41 23 the energies of the stationary states for a particle in a box are En n2E1 where En is the energy of the stationary state with quantum number n It can be seen either from Figure 41 7 or from the that the wave function given in Figure Ex41 4 corresponds to n wave function equation n x L sin A n x 4 Thus E 4 16 E E 1 E 4 16 12 0 eV 16 0 75 eV 41 5 Solve From Equation 41 41 the units of the penetration distance are cid 61 2 m U E 0 s J kg J 2 2 kg m s kg kg m s 2 s 2 2 kg m s 2 2 kg m s 2 2 kg m s kg m s m 41 6 Solve a b For n 2 the probability of finding the particle at the center of the well is zero This is because the wave function is zero at that point c This is consistent with standing waves The n 2 standing wave on a string has a node at the center of the string 41 7 Model The wave function decreases exponentially in the classically forbidden region Solve The probability of finding a particle in the small interval x at position x is Prob in x at x Thus the ratio 2 x x Prob in at x x Prob in x L x L at 2 2 L 2 2 L L x L x The wave function in the classically forbidden region x L is x L x e edge At the edge of the forbidden region at x L L edge At x L L edgee 1 Thus Prob in e x at Prob in x x L x L at 2 L 2 L 1 2 2 edge edge 2 e 0 135 41 8 Solve a According to Equation 41 41 the penetration distance is cid 61 1 05 10 34 J s 2 m U E 0 2 9 11 10 kg 2 0 eV 0 5 eV 1 60 10 31 19 J eV 0 159 nm b Likewise for E 1 00 eV c For E 1 50 eV 0 275 nm Assess These values are of the correct order of magnitude as you can see by referring to Figure 41 14 a 0 195 nm 41 9 Solve According to Equation 41 41 the penetration depth is cid 61 2m U E 0 Hence U E 0 2 cid 61 2 m 2 34 1 05 10 J s 2 2 9 11 10 kg 1 0 10 m 9 31 2 6 05 10 J 21 0 038 eV 1 eV 1 6 10 19 J The electron s energy is 0 038 eV below 0 U 41 10 Solve According to Equation 41 41 the penetration depth is cid 61 2 m U E 0 2 4 1 661 10 kg 1 0 eV 1 60 10 19 J eV 34 J s 1 05 10 27 2 28 10 12 m 2 28 pm 41 11 Visualize Solve There are three factors to consider First the de Broglie wavelength increases as the particle s speed and kinetic energy decreases Thus the spacing between the nodes of x increases in regions where U is larger Second a particle is more likely to be found where it is moving the slowest Thus the amplitude of x increases in regions where U is larger Third for n 6 there will be six antinodes to place 41 12 Visualize Solve There are three factors to consider First the de Broglie wavelength increases as the particle s speed and kinetic energy decreases Thus the spacing between the nodes of x increases in regions where U is larger Second a particle is more likely to be found where it is moving the slowest Thus the amplitude of x increases in regions where U is larger Third for n 8 there will be eight antinodes to place 41 13 Visualize a The energy diagram is shown above Solve b There are three factors to consider First the de Broglie wavelength increases as the particle s speed and kinetic energy decreases Thus the spacing between the nodes of x increases in regions where U is larger Second a particle is more likely to be found where it is moving the slowest Thus the amplitude of x increases in regions where U is larger Third n 3 has 3 antinodes and n 6 has six antinodes 41 14 Visualize The steps of Tactics Box 41 1 have been followed to sketch the wave functions shown in the figure 41 15 Model The electron is a quantum harmonic oscillator Solve Using Equation 41 48 for the energy levels of the electron the energy of the photon emitted in the 3 2 and 3 1 quantum jumps are E 3 E 2 3 cid 61 e 2 cid 61 e cid 61 …
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