Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Chapter 10, Problem 10.81E
Interpretation Introduction

Interpretation:

The degeneracies for all energies of a 2-D particle-in-a-box up to nx=ny=8 are to be stated. The plot of the energy levels in a graph is to be drawn.

Concept introduction:

The Schrodinger equation is used to find the allowed energy levels for electronic transitions in the quantum mechanics. It is generally expressed as shown below.

HΨ=EΨ

Where,

H is the Hamiltonian operator.

Ψ is the wavefunction.

E is the energy.

The energy obtained after applying the operator on wavefunction is known as the eigen value for the wavefunction.

Expert Solution & Answer
Check Mark

Answer to Problem 10.81E

The degeneracies for all energies of a 2-D particle-in-a-box up to nx=ny=8 have been calculated and the plot of energy versus energy levels in a graph is shown below.

Physical Chemistry, Chapter 10, Problem 10.81E , additional homework tip  1

Explanation of Solution

The formula to calculate energy for 2-D box can be written as given below.

E=h28m(nx2a2+ny2b2)

Assuming a=b, the energy corresponding to lowest possible quantum numbers (1,1) are calculated as shown below.

E1,1=h28m(nx2a2+ny2b2)=h28m(12a2+12a2)=h28m(1a2+1a2)=2h28ma2

Similarly, energies corresponding to energy levels (1,2)(2,1) are calculated as shown below.

E1,2=5h28ma2E2,1=5h28ma2

Similarly, energies corresponding to energy levels (1,3)(3,1) are calculated as shown below.

E1,3=10h28ma2E3,1=10h28ma2

Similarly, energies corresponding to energy levels (1,4)(4,1) are calculated as shown below.

E1,4=17h28ma2E4,1=17h28ma2

Similarly, energies corresponding to energy levels (2,3)(3,2) are calculated as shown below.

E2,3=13h28ma2E3,2=13h28ma2

Similarly, energies corresponding to energy levels (2,4)(4,2) are calculated as shown below.

E2,4=20h28ma2E4,2=20h28ma2

Similarly, energies corresponding to energy levels (3,4)(4,3) are calculated as shown below.

E3,4=25h28ma2E4,3=25h28ma2

Similarly, energies corresponding to energy levels (1,5)(5,1) are calculated as shown below.

E1,5=26h28ma2E5,1=26h28ma2

Similarly, energies corresponding to energy levels (2,5)(5,2) are calculated as shown below.

E2,5=29h28ma2E5,2=29h28ma2

Similarly, energies corresponding to energy levels (3,5)(5,3) are calculated as shown below.

E3,5=34h28ma2E5,3=34h28ma2

Similarly, energies corresponding to energy levels (1,6)(6,1) are calculated as shown below.

E1,6=37h28ma2E6,1=37h28ma2

Similarly, energies corresponding to energy levels (2,6)(6,2) are calculated as shown below.

E2,6=40h28ma2E6,2=40h28ma2

Similarly, energies corresponding to energy levels (4,5)(5,4) are calculated as shown below.

E4,5=41h28ma2E5,4=41h28ma2

Similarly, energies corresponding to energy levels (3,6)(6,3) are calculated as shown below.

E3,6=45h28ma2E6,3=45h28ma2

Similarly, energies corresponding to energy levels (1,7)(7,1)(5,5) are calculated as shown below.

E1,7=50h28ma2E7,1=50h28ma2E5,5=50h28ma2

Similarly, energies corresponding to energy levels (4,6)(6,4) are calculated as shown below.

E4,6=52h28ma2E6,4=52h28ma2

Similarly, energies corresponding to energy levels (2,7)(7,2) are calculated as shown below.

E2,7=53h28ma2E7,2=53h28ma2

Similarly, energies corresponding to energy levels (3,7)(7,3) are calculated as shown below.

E3,7=58h28ma2E7,3=58h28ma2

Similarly, energies corresponding to energy levels (6,5)(5,6) are calculated as shown below.

E6,5=61h28ma2E5,6=61h28ma2

Similarly, energies corresponding to energy levels (1,8)(8,1)(4,7)(7,4) are calculated as shown below.

E1,8=65h28ma2E8,1=65h28ma2E4,7=65h28ma2E7,4=65h28ma2

Similarly, energies corresponding to energy levels (2,8)(8,2)are calculated as shown below.

E2,8=68h28ma2E8,2=68h28ma2

Similarly, energies corresponding to energy levels (3,8)(8,3) are calculated as shown below.

E3,8=73h28ma2E8,3=73h28ma2

Similarly, energies corresponding to energy levels (5,7)(7,5)are calculated as shown below.

E5,7=74h28ma2E7,5=74h28ma2

Similarly, energies corresponding to energy levels (4,8)(8,4) are calculated as shown below.

E4,8=80h28ma2E8,4=80h28ma2

Similarly, energies corresponding to energy levels (6,7)(7,6) are calculated as shown below.

E6,7=85h28ma2E7,6=85h28ma2

Similarly, energies corresponding to energy levels (5,8)(8,5) are calculated as shown below.

E5,8=89h28ma2E8,5=89h28ma2

Similarly, energies corresponding to energy levels (6,8)(8,6) are calculated as shown below.

E6,8=100h28ma2E8,6=100h28ma2

Similarly, energies corresponding to energy levels (7,8)(8,7) are calculated as shown below.

E7,8=113h28ma2E8,7=113h28ma2

Similarly, energy corresponding to energy levels (2,2) is calculated as shown below.

E2,2=8h28ma2

Similarly, energy corresponding to energy levels (3,3) is calculated as shown below.

E3,3=18h28ma2

Similarly, energy corresponding to energy levels (4,4) is calculated as shown below.

E4,4=32h28ma2

Similarly, energy corresponding to energy levels (6,6) is calculated as shown below.

E6,6=72h28ma2

Similarly, energy corresponding to energy levels (7,7) is calculated as shown below.

E7,7=98h28ma2

Similarly, energy corresponding to energy levels (8,8) is calculated as shown below.

E8,8=128h28ma2

The plot of energy versus energy levels is shown in the figure 1.

Physical Chemistry, Chapter 10, Problem 10.81E , additional homework tip  2

Figure 1

Conclusion

The degeneracies for all energies of a 2-D particle-in-a-box up to nx=ny=8 have been calculated and the plot of energy versus energy levels is shown in figure 1.

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