
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
- A student proposes the transformation below in one step of an organic synthesis. There may be one or more reactants missing from the left-hand side, but there are no products missing from the right-hand side. There may also be catalysts, small inorganic reagents, and other important reaction conditions missing from the arrow. • Is the student's transformation possible? If not, check the box under the drawing area. . If the student's transformation is possible, then complete the reaction by adding any missing reactants to the left-hand side, and adding required catalysts, inorganic reagents, or other important reaction conditions above and below the arrow. • You do not need to balance the reaction, but be sure every important organic reactant or product is shown. + T X O O лет-ле HO OH HO OH This transformation can't be done in one step.arrow_forwardDetermine the structures of the missing organic molecules in the following reaction: X+H₂O H* H+ Y OH OH Note: Molecules that share the same letter have the exact same structure. In the drawing area below, draw the skeletal ("line") structures of the missing organic molecules X and Y. You may draw the structures in any arrangement that you like, so long as they aren't touching. Click and drag to start drawing a structure. X Sarrow_forwardPredict the major products of this organic reaction. If there aren't any products, because nothing will happen, check the box under the drawing area instead. No reaction. HO. O :☐ + G Na O.H Click and drag to start drawing a structure. XS xs H₂Oarrow_forward
- What are the angles a and b in the actual molecule of which this is a Lewis structure? H H C H- a -H b H Note for advanced students: give the ideal angles, and don't worry about small differences from the ideal groups may have slightly different sizes. a = b = 0 °arrow_forwardWhat are the angles a and b in the actual molecule of which this is a Lewis structure? :0: HCOH a Note for advanced students: give the ideal angles, and don't worry about small differences from the ideal that might be caused by the fact that different electron groups may have slightly different sizes. a = 0 b=0° Sarrow_forwardDetermine the structures of the missing organic molecules in the following reaction: + H₂O +H OH O OH +H OH X Note: Molecules that share the same letter have the exact same structure. In the drawing area below, draw the skeletal ("line") structure of the missing organic molecule X. Click and drag to start drawing a structure.arrow_forward
- Identify the missing organic reactant in the following reaction: x + x O OH H* + ☑- X H+ O O Х Note: This chemical equation only focuses on the important organic molecules in the reaction. Additional inorganic or small-molecule reactants or products (like H₂O) are not shown. In the drawing area below, draw the skeletal ("line") structure of the missing organic reactant X. Click and drag to start drawing a structure. Carrow_forwardCH3O OH OH O hemiacetal O acetal O neither O 0 O hemiacetal acetal neither OH hemiacetal O acetal O neither CH2 O-CH2-CH3 CH3-C-OH O hemiacetal O acetal CH3-CH2-CH2-0-c-O-CH2-CH2-CH3 O neither HO-CH2 ? 000 Ar Barrow_forwardWhat would be the best choices for the missing reagents 1 and 3 in this synthesis? 1. PPh3 2 2. n-BuLi 3 Draw the missing reagents in the drawing area below. You can draw them in any arrangement you like. • Do not draw the missing reagent 2. If you draw 1 correctly, we'll know what it is. • Note: if one of your reagents needs to contain a halogen, use bromine. Explanation Check Click and drag to start drawing a structure.arrow_forward
- Predict the products of this organic reaction: NaBH3CN + NH2 ? H+ Click and drag to start drawing a structure. ×arrow_forwardPredict the organic products that form in the reaction below: + OH +H H+ ➤ ☑ X - Y Note: You may assume you have an excess of either reactant if the reaction requires more than one of those molecules to form the products. In the drawing area below, draw the skeletal ("line") structures of the missing organic products X and Y. You may draw the structures in any arrangement that you like, so long as they aren't touching. Click and drag to start drawing a structure. Garrow_forwardPredict the organic products that form in the reaction below: OH H+ H+ + ☑ Y Note: You may assume you have an excess of either reactant if the reaction requires more than one of those molecules to form the products. In the drawing area below, draw the skeletal ("line") structures of the missing organic products X and Y. You may draw the structures in any arrangement that you like, so long as they aren't touching. Click and drag to start drawing a structure. ✓ marrow_forward
- Physical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,Chemistry: Principles and PracticeChemistryISBN:9780534420123Author:Daniel L. Reger, Scott R. Goode, David W. Ball, Edward MercerPublisher:Cengage Learning

