Mueller M.R. - Fundamentals of Quantum Chemistry[c] Molecular Spectroscopy and Modern Electronic Structure Computations (Kluwer, 2001)

2.2) For a Particle in a 1-dimensional Box, determine whether the following operators are Hermitian: a) position; b) momentum; and c) kinetic energy.

2.3) Demonstrate that the eigenfunctions for the Particle in a 1- dimensional Box are orthonormal.

2.4) For the particle in a 1-dimensional Particle-in-a-Box, determine the probability of the particle in the center of the box for the and states. Justify your results based on the shapes of the wavefunctions in this region as shown in Figure 2-2. How does this compare to what would be predicted classically?

2.5) For the Particle in a 1-dimensional Box, what is the probability that the particle is at in the and states?

2.6) Determine the energy eigenvalues and eigenfunctions for a particle free to travel from to What occurs to the quantization of the particle’s energy? What might you infer about the curvature of the eigenfunctions?

2.7) Calculate explicitly the momentum expectation value for the 1- dimensional Particle-in-a-Box for the state.

2.8) Calculate the uncertainty in the momentum and position for a 1- dimensional Particle-in-a-Box for the and states. Is the product of the uncertainties within Heisenberg Uncertainty Principle limits?

2.9) Determine the uncertainty in the momentum and position for each dimension for a 3-dimensional Particle-in-a-Box in the ground-state. Why is it not possible for the energy to be zero in any one dimension even if it is not in the other dimensions?

2.10) Consider a particle in a 3-dimensional box with the following dimensions: Does this system have any degenerate states? Justify your answer.

The second derivatives with respect to x and y in Equation 3-1 are transformed into polar coordinates using the chain rule.

The Hamiltonian in Equation 3-1 can now be written in terms of r and

Since the radius r is constant in this problem, all of the terms that involve derivatives with respect to r will be zero, reducing the Hamiltonian to just one variable,

The moment of inertia, I, of the particle, is equal to its mass times the square of the radius of gyration r. The Hamiltonian in Equation 3-2 is very similar to the Hamiltonian for the one-dimensional Particle-in-a-Box problem (see

Section 2.4). This will result in the same functional forms for the wavefunctions in terms of the variable The following wavefunction will be used:

The constants A and B are normalization constants, and the constant k will be determined by the boundary conditions for the particle. The Schroedinger equation and the energy eigenvalue becomes:

The boundary conditions must now be applied to the system. There are no points along the circular path where the potential becomes infinite as in the case of the Particle-in-a-Box; hence, the wavefunction does not need to truncate at any points. The wavefunction, however, must be continuous and single valued at a given point (see Section 2.3). This requires that the value of the wavefunction must be the same at and

The equality in Equation 3-5 is satisfied only if k is an integer (since then and individually are equal to 1). The symbol for k is traditionally known as the magnetic quantum number when referring to electronic states. The possible values for are as follows:

The allowed energy values become:

The final task is to normalize the wavefunction. Since the value of can be both positive and negative integers, the wavefunction can be reduced to just one exponential term.

The limits of integration for normalization will be from 0 to since this covers the entire circular path.

The normalized wavefunction for the Particle-on-a-Ring becomes:

A physical connection needs to be made of the sign on the quantum states. Positive and negative signs indicate direction, and in this case, the direction must be the direction of rotation: clockwise or counterclockwise. To confirm this, the angular momentum, L, of the particle can be determined. Since the rotation is confined to the x-y plane, the only nonzero component of the angular momentum of the particle is along the z-axis,

The angular momentum operator along the z-axis in polar coordinates is given as:

Based on the left-hand rule, positive angular momentum (for positive values of ) indicates a clockwise rotation whereas negative values indicate a counterclockwise rotation.

It is interesting to note that in this system, the ground-state energy

is zero unlike in the Particle-in-a-Box system. There are no points along the circular path that require the wavefunction to become zero; hence, it is possible for the wavefunction to have no curvature. As discussed previously in the Particle-in-a-Box system (see Section 2.4), the degree of curvature in a wavefunction is related to the amount of kinetic energy that the particle possesses. Since the ground-state wavefunction is a constant, there is no curvature to the wavefunction and the particle has no kinetic energy or correspondingly angular momentum. The probability density of the particle in the ground-state is the same throughout the entire circular path since the wavefunction is a standing wave of constant amplitude. The real portion of the wavefunctions for the states of +1 and +2 are shown in Figures 3-2a and 3-2b. The states of –1 and –2 are same as shown Figure 3-2 except the waves are now inverted. In these higher states, curvature in the wavefunctions emerges indicating a non-zero kinetic energy. The anti-nodes of the wavefunction lie above and below the ring.

Since Equation 3-15 is a sum of derivatives with respect to and it is separable. As a result, the wavefunction will be a product of wavefunctions in terms of and as previously discussed in Section 2.7.

The first part of the Schroedinger equation in Equation 3-15 involves taking the second derivative of the wavefunction with respect to This is identical to the operation as seen previously for the Particle-on-a-Ring; hence, the wavefunction will be the same as in the Particle-on-a-Ring.

Substitution of Equation 3-16 into Equation 3-15 results in the following differential equation:

The solution to this differential equation is well known. The wavefunctions that satisfy Equation 3-17 are the associated Legendre polynomials. The associated Legendre polynomials, for a variable z are obtained from the following recursion relationship:

The variable for the function is actually cos hence, the associated Legendre polynomials are generated in terms of z, and then z is replaced by cos throughout. The normalization constant is obtained by integrating from to

The normalized associated Legendre polynomials, are given in the following expression, and the first several are listed in Table 3-1.

The product of the normalized associated Legendre polynomials along with the Particle-on-a-Ring functions are known as the spherical harmonics symbolized as

The arbitrary phase factor

is introduced to conform to common conventions and its value is +1 or –1. Table 3-1 lists the first few spherical harmonic wavefunctions. When the legendrian, is applied to a spherical harmonic wavefunction, the following eigenvalue results: