Concept explainers
(a)
Interpretation:
The eigenvalues of total
Concept introduction:
The eigenvalues of the wavefunction that are obtained when an operator is applied are the only possible values of observables. The expression for the eigenvalue is given by,
The total angular momentum does not depend on the mass of the particle, radius of the rotation and also the magnetic quantum number.
Answer to Problem 11.54E
The eigenvalues of total angular momentum is
Explanation of Solution
Explanation:
The general equation for the wavefunction in the 3-dimensional rotation is,
The complete form of
The total angular momentum using the complete forms of operators is,
The first derivative of the given wavefunction with respect to
The second derivative of the given wavefunction with respect to
The second derivative of the given wavefunction with respect to
Substitute equation (1), (2) and (3) in the equation of total angular momentum as shown below.
Take common terms together and rearrange the given equation as shown below.
Substitute the value of
Substitute
Thus, the total angular momentum is represented as,
The eigenvalues of total angular momentum is
The eigenvalues of total angular momentum is
(b)
Interpretation:
The eigenvalues of z-component of angular momentum is to be evaluated using the complete forms of given wavefunction
Concept introduction:
The eigenvalues of the wavefunction that are obtained when an operator is applied are the only possible values of observables. The expression for the eigenvalue is given by,
The z-component of the three dimensional angular momentum that has components in x, y and z direction is quantized.
Answer to Problem 11.54E
Explanation of Solution
The general equation for the wavefunction in the 3-dimensional rotation is,
The complete form of
The z-component of angular momentum using the complete forms of operators is,
The first derivative of the given wavefunction with respect to
Substitute equation (4) in the equation of z-component of angular momentum as shown below.
The eigenvalues of z-component of angular momentum is
The eigenvalues of z-component of angular momentum is
(c)
Interpretation:
The eigenvalue of energy is to be evaluated using the complete forms of given wavefunction
Concept introduction:
The eigenvalues of the wavefunction that are obtained when an operator is applied are the only possible values of observables. The expression for the eigenvalue is given by,
The energy of the particle depends on the moment of inertia, quantum number and Planck’s constant. The total energy is quantized.
Answer to Problem 11.54E
The eigenvalue of energy for the given wavefunctionis
Explanation of Solution
The general equation for the wavefunction in the 3-dimensional rotation is,
The complete form of
The eigen equation for the Hamiltonian operator is,
The Hamiltonian operator for energy applied on the given wavefunction is also represented in the form of total angular momentum.
The value of total angular momentum is
The eigenvalue of energy
The eigenvalue of energy
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Chapter 11 Solutions
Physical Chemistry
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