Modern Physics, 3rd Edition
Modern Physics, 3rd Edition
3rd Edition
ISBN: 9780534493394
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
Question
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Chapter 9, Problem 17P

(a)

To determine

The lowest energy of the system in which electrons occupy respective state.

(a)

Expert Solution
Check Mark

Answer to Problem 17P

The lowest energy of the system in which electrons occupy respective state is 394.8eV.

Explanation of Solution

The particle placed in a cubical box is a well-known model in the field of quantum mechanics which illustrates the motion of a particle confined to move inside a 3 dimensional box. The walls of this cubical box are impenetrable as the potential of these walls is infinite.

The particle in a box is a completely hypothetical model which illustrates the basic difference between the classical and quantum models. According to classical mechanics, the particle confined in a box can move with any velocity whereas in quantum mechanics the particles can occupy only particular states due to the presence of quantum effects.

Write the expression for the energy of the particle in cubical box.

  E=2π22mL2(n12+n22+n32)        (I)

Here, E is the energy of the system, L is the side of the cubical box, m is the mass of the confined particle and n is the state occupied by the particle.

According to Pauli Exclusion Principle, no more than two fermions can occupy same state. The electros are also fermions and are identical due to which any two electrons can occupy any state.

The electrons will, first, occupy ground state and then they will occupy further states with two electrons filled in each state.

The minimum energy of the system is the sum of energies of electrons present in ground state and other states. The electrons can also occupy the degenerate energy state due to which there can be three possible combinations, of respective n1, n2 and n3, (1,1,2),(1,2,1) and (2,1,1).

The energy of the electrons presents in the states (1,1,2),(1,2,1) and (2,1,1) is equal.

Write the expression for the minimum energy of the system of 8 electrons.

  Emin=2(E111+E112+E121+E211)

Simplify the above expression.

  Emin=2(E111+3E112)        (II)

Here, Emin is the minimum energy of the system, E111 is the energy of electrons present in ground state and E112 is the energy of the electrons present in (1,1,2),(1,2,1) and (2,1,1).

Conclusion:

Substitute E111 for E, 1.054×1034Js for , 9.11×1031kg for m, 0.2nm for L,1 for n1, 1 for n2 and 1 for n3 in equation (I).

  E111=(1.054×1034Js)2π22(9.11×1031kg)(0.2nm(109m1nm))2(12+12+12)=4.5×1018J(1eV1.6×1019J)=28.2eV

Substitute E112 for E, 1.054×1034Js for , 9.11×1031kg for m, 0.2nm for L,1 for n1, 1 for n2 and 2 for n3 in equation (I).

  E112=(1.054×1034Js)2π22(9.11×1031kg)(0.2nm(109m1nm))2(12+12+22)=9×1018J(1eV1.6×1019J)=56.4eV

Substitute 56.4eV for E112 and 28.2eV for E111 in equation (II).

  Emin=2(28.2eV+3(56.4eV))=394.8eV

Thus, the lowest energy of the system in which electrons occupy respective state is 394.8eV.

(b)

To determine

The lowest energy of the system of particles which have same mass as electrons but do not obey exclusion principle.

(b)

Expert Solution
Check Mark

Answer to Problem 17P

The lowest energy of the system of particles which have same mass as electrons but do not obey exclusion principle is 225.6eV.

Explanation of Solution

Since, the particles do not obey exclusion principle. Therefore all the particles can occupy same state that is ground state.

Write the expression for the minimum energy.

  Emin=8E111        (III)

Here, Emin is the minimum energy of the system of particles present in the same state.

Conclusion:

Substitute E111 for E, 1.054×1034Js for , 9.11×1031kg for m, 0.2nm for L,1 for n1, 1 for n2 and 1 for n3 in equation (I).

  E111=(1.054×1034Js)2π22(9.11×1031kg)(0.2nm(109m1nm))2(12+12+12)=4.5×1018J(1eV1.6×1019J)=28.2eV

Substitute 28.2eV for E111 in equation (III).

  Emin=8(28.2eV)=225.6eV

Thus, the lowest energy of the system of particles which have same mass as electrons but do not obey exclusion principle is 225.6eV.

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