Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Question
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Chapter 28, Problem 70P

(a)

To determine

The length of the one-dimensional box.

(a)

Expert Solution
Check Mark

Answer to Problem 70P

The length of the box is 6.1 nm.

Explanation of Solution

The ground state energy of the electron in a box is 0.010 eV.

Write the expression for energy of the quantized level.

En=n2h28mL2                                                            (I)

Here, En is the energy of the level n, n is the quantum number, h is the Planck’s constant, m is the mass of the electron and L is the length of the box.

Write the expression for ground state energy for a particle in box.

E1=h28mL2                                                           (II)

Here, E1 is the ground state energy.

Rearranging (I)

L=h28mE1                                                            (III)

Substituting 9.109×1031 kg for m, 0.010 eV for E1 and 6.626×1034 Js for h in (II) to find L.

 L=(6.626×1034 Js)28×9.109×1031 kg×0.010 eV=(6.626×1034 Js)28×9.109×1031 kg×0.010 eV×1 eV1.602×1019J=6.1×109m6.1nm

Thus, the length of the box is 6.1 nm.

(b)

To determine

The plot of wave functions of the first three state.

(b)

Expert Solution
Check Mark

Explanation of Solution

Write the expression for the wave function of a particle in a box.

ψn=2Lsin(nπxL)                                                     (IV)

Here, ψn is the wave function of the quantum state n, L is the length of the box  and x is the position co-ordinate.

Substitute 1,2 and 3 for n and 6.1 nm for L  in (VIII) to find the fourth excited state wave function, ψ5.

ψ1=26.1 nmsin(πx6.1 nm)                                                    (V)

Here, ψ1 is the ground state wave function.

ψ2=26.1 nmsin(2πx6.1 nm)                                                    (VI)

Here, ψ2 is the ground state wave function.

ψ3=26.1 nmsin(3πx6.1 nm)                                                    (VII)

Here, ψ3 is the ground state wave function.

The plot of the wave functions versus position is given below:

Physics, Chapter 28, Problem 70P , additional homework tip  1

Physics, Chapter 28, Problem 70P , additional homework tip  2

Physics, Chapter 28, Problem 70P , additional homework tip  3

(c)

To determine

The wavelength of the electron in the third state.

(c)

Expert Solution
Check Mark

Answer to Problem 70P

The wave length of the electron in the second excited state is 4.1 nm.

Explanation of Solution

Substituting 3 for n in (III)

E3=32h28mL2                                                           (IV)

Here, E3 is the energy of the second excited state.

Write the expression for de Broglie wavelength of an electron.

λ=h2mK                                                                     (V)

Here, λ is the de Broglie wavelength of the electron and K is the kinetic energy of the electron.

The particle in a box model renders only the kinetic energy since the potential is zero at the box and is infinite everywhere else. Thus, consider, K=E3.

Substituting E3 for K  in (V)

λ=h2mK8mL29h2                                                                     (VI)

Rearranging

λ=2L3                                                                   (VII)

Substituting 6.1nm for L in (VII) to find λ.

λ=2×6.1 nm3=4.1 nm

Thus, the wave length of the electron in the second excited state is 4.1 nm.

(d)

To determine

The wavelength of the emitted photon during the downward transition of the electron.

(d)

Expert Solution
Check Mark

Answer to Problem 70P

The possible wavelengths of the emitted photons are 15.5 μm, 25 μm and41 μm.

Explanation of Solution

The wavelength of the photon is 15.5 μm.

Write the expression for energy of a photon.

E=hcλ                                                                    (IX)

Here, E is the energy of the photon, λ is the wavelength of the photon  and c is the speed of light.

Write the expression for energy of the quantized levels in terms of the ground state energy

En=n2E1                                                                     (X)

Subtracting E1 from (X)

 EnE1=n2E1E1                                                                     (XI)

Here, EnE1 is the energy absorbed by the electron from the photon to undergo an upward transition. Thus, EnE1 is same as E and hence (X) becomes

E=(n21)E1                                                                     (XII)

Rearranging (XII)

n=EE1+1                                                                 (XIII)

Substituting 15.5 μm for λ, 1240 eV for hc in (IX) to find E

E=1240eVnm15.5 μm=0.0800 eV

Substitute 0.0800 eV for E and  0.010 eV for E1 in (XII) to find n

n=0.0800 eV0.010 eV+1=3

Thus when the electron absorbs the photon, it is excited to the third state.

The possible downward transitions are 32,21and 31.

Write the expression for energy of the photon during a downward transition.

λ=hcEnEn                                                               (XIV)

Here, En is the energy in the higher state and En is the energy in the lower state.

Substituting 2and 3 for n in (X) to find E2and E3 respectively

E2=22E1                                                          (XV)

E3=32E1                                                          (XVI)

Substituting 2 for n and 1 for n in (XIV) to find λ

λ=hcE2E1                                                     (XVII)

Substituting 1240 eV for hc, 22E1 for E2 and 0.010 eV for E1 in (XVII) to find λ

λ=1240 eVnm4(0.010 eV)0.010 eV=41×106 m=41 μm

For, 21 transition, the wavelength of the photon is 41 μm.

Substituting 3 for n and 2 for n in (XIV) to find λ

λ=hcE3E2                                                    (XVIII)

Substituting 1240 eV for hc, 32E1 for E3, 22E1 for E2 and 0.010 eV for E1 in (XVIII) to find λ

λ=1240 eVnm9(0.010 eV)4(0.010 eV)=25×106 m=25μm

For, 32 transition, the wavelength of the photon is 25μm.

For, 31 transition, the emitted photon has the same energy as the absorbed photon. Thus, the wavelength of the photon is 15.5μm.

Thus, the possible wavelengths of the emitted photons are 15.5 μm, 25 μm and41 μm.

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Chapter 28 Solutions

Physics

Ch. 28 - Prob. 2CQCh. 28 - Prob. 3CQCh. 28 - Prob. 4CQCh. 28 - Prob. 5CQCh. 28 - Prob. 6CQCh. 28 - Prob. 7CQCh. 28 - Prob. 8CQCh. 28 - Prob. 9CQCh. 28 - Prob. 10CQCh. 28 - Prob. 11CQCh. 28 - Prob. 12CQCh. 28 - Prob. 13CQCh. 28 - Prob. 14CQCh. 28 - Prob. 15CQCh. 28 - Prob. 16CQCh. 28 - Prob. 17CQCh. 28 - Prob. 18CQCh. 28 - Prob. 1MCQCh. 28 - Prob. 2MCQCh. 28 - Prob. 3MCQCh. 28 - Prob. 4MCQCh. 28 - Prob. 5MCQCh. 28 - Prob. 6MCQCh. 28 - Prob. 7MCQCh. 28 - Prob. 8MCQCh. 28 - Prob. 9MCQCh. 28 - Prob. 10MCQCh. 28 - Prob. 1PCh. 28 - Prob. 2PCh. 28 - Prob. 3PCh. 28 - Prob. 4PCh. 28 - Prob. 5PCh. 28 - Prob. 6PCh. 28 - Prob. 7PCh. 28 - Prob. 8PCh. 28 - Prob. 9PCh. 28 - Prob. 10PCh. 28 - Prob. 11PCh. 28 - Prob. 12PCh. 28 - Prob. 13PCh. 28 - Prob. 15PCh. 28 - Prob. 14PCh. 28 - Prob. 17PCh. 28 - Prob. 16PCh. 28 - Prob. 18PCh. 28 - Prob. 19PCh. 28 - Prob. 20PCh. 28 - Prob. 21PCh. 28 - Prob. 23PCh. 28 - Prob. 22PCh. 28 - Prob. 25PCh. 28 - Prob. 24PCh. 28 - Prob. 26PCh. 28 - Prob. 27PCh. 28 - Prob. 28PCh. 28 - Prob. 29PCh. 28 - Prob. 30PCh. 28 - Prob. 32PCh. 28 - Prob. 31PCh. 28 - Prob. 33PCh. 28 - Prob. 34PCh. 28 - Prob. 35PCh. 28 - Prob. 36PCh. 28 - Prob. 37PCh. 28 - Prob. 39PCh. 28 - Prob. 41PCh. 28 - Prob. 40PCh. 28 - Prob. 38PCh. 28 - Prob. 42PCh. 28 - Prob. 43PCh. 28 - Prob. 44PCh. 28 - Prob. 45PCh. 28 - Prob. 46PCh. 28 - Prob. 47PCh. 28 - Prob. 48PCh. 28 - Prob. 49PCh. 28 - Prob. 50PCh. 28 - Prob. 51PCh. 28 - Prob. 52PCh. 28 - Prob. 53PCh. 28 - Prob. 54PCh. 28 - Prob. 55PCh. 28 - Prob. 56PCh. 28 - Prob. 57PCh. 28 - Prob. 58PCh. 28 - Prob. 59PCh. 28 - Prob. 60PCh. 28 - Prob. 61PCh. 28 - Prob. 62PCh. 28 - Prob. 63PCh. 28 - Prob. 65PCh. 28 - Prob. 64PCh. 28 - Prob. 66PCh. 28 - Prob. 67PCh. 28 - Prob. 68PCh. 28 - Prob. 69PCh. 28 - Prob. 70PCh. 28 - Prob. 71PCh. 28 - Prob. 72PCh. 28 - Prob. 73PCh. 28 - Prob. 74PCh. 28 - Prob. 75PCh. 28 - Prob. 76PCh. 28 - Prob. 77PCh. 28 - Prob. 79PCh. 28 - Prob. 78PCh. 28 - Prob. 80PCh. 28 - Prob. 81PCh. 28 - Prob. 82PCh. 28 - Prob. 83PCh. 28 - Prob. 84P
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