Module 12:2-D and 3-D Quantum Problems
I.Particle In A 2-D Box With Infinite Walls
Consider a particle trapped in a 2-D infinite box as shown below:
The Time Independent Schrodinger Equation for a multi-dimensional problem is given by
where .
Using the Cartesian form of the Laplacian, we have inside the box
where
and that is zero outside the box.
Since each term of the potential energy depends on at most one spatial coordinate (in this case no coordinates), we can use separation of variables to simplify our work. Thus, we choose a product solution for the wave function where X and Y are each functions of only one spatial coordinate.
Substituting into the Time Independent Schrodinger Equation, we have
Each term depends on different spatial coordinates which are independent. Since the equation must hold for any point inside the box, each term must be a constant!
Furthermore these constants must have the dimensions of inverse square meters (i.e. wave number squared). Thus, it makes sense to write the individual constants in a manner similar to what we did with the 1-D well as follows:
We also note the additional requirement:
This is just the dot product of a vector (wave number) upon itself where
.
From our work on the 1-D infinite square well, we know the general solutions to these two equations are given by
where 0 x L
where 0 yb
Thus, the energy wave function is
The boundary condition that is zero for all values of y when x = 0 requires that B = 0. The boundary condition that is zero for all values of x when y = 0 requires that E = 0. The boundary condition that that is zero for all values of y when x = L requires that where n = 1, 2, 3, … The boundary condition that that is zero for all values of x when y = b requires that where q = 1, 2, 3, … By plugging the results for the components of the wave number into our energy equations, we obtain the energy eigenvalues:
where
To find the normalization constants, we integrate over all space in the box:
.
Although only the product of the constants A and D has physical meaning, it is often common to normalize each individual function X and Y independently. In this manner, you can think of the problem as simply being the product two 1-D infinite well problems from Module 10. Doing this gives us, the following:
Thus, the energy eigenfunctions for this problem are
with associated energy eigenvalues of
where
Since this is a 2-dimensional problem, there must be at least two quantum numbers which determine an energy state’s wave function and energy eigenvalue (one for each spatial coordinate). This is an important concept and has led us to find additional variables which effect a systems energy but which are not spatial in nature like spin!! It is also possible for two different energy wave functions (energy levels) to have the same energy eigenvalues. This is called degeneracy. One way in which this might happen is if the system possesses some symmetry. For instance if the box is a square with sides of length L then E1,3 and E3,1 state would have the same energy as there is no difference to the system is you exchange the x and y coordinates (rotate 90 degrees). This is called symmetrical degeneracy. Many of the excited energy levels in the square system would be degenerate. In other cases, degeneracy might occur just due to pure luck. This is called accidental degeneracy. For instance, our original system is degenerate for any group of (n,q) pairs where .
II.Hydrogen Atom
A. Coulomb Potential
To solve for any quantum problem, we first must find the potential energy function for the problem. In the case of the hydrogen atom or any single electron ion, the potential energy function is the Coulomb potential which can be written in Cartesian coordinates as
where . Although it is technically correct to write the Coulomb potential in Cartesian coordinates, it is a poor choice for solving the Schrodinger equation since:
1) We can’t use separation of variable when the potential has terms containing more than one variable.
2)The system and hence its boundary conditions has spherical symmetry.
Hence, we need to write the Coulomb potential using spherical coordinates as
where Z = 1 for Hydrogen.
B.Reduced Mass -
In order to reduce the hydrogen atom from a two body problem to a one body problem, we must replace the mass of the electron with the reduced mass (as previously discussed in the module on the Bohr atom)
where M is the mass of the proton and m is the mass of the electron.
C.Laplacian in Spherical Coordinates
The Laplacian is much more complicated in coordinate systems other than Cartesian, but it is necessary to use these coordinate systems for reasons previously discussed. The Laplacian in various coordinate systems can be found in math handbooks like CRC or Schaum’s. The Laplacian in spherical coordinate systems is given as
D.Time Independent Schrodinger Equation
We can now put everything together to write the T.I.S.E. for the hydrogen atom.
We now write the wavefunction as a product of three single variable functions as
.
Substituting into the T.I.S.E., we have
We can isolate the phi dependence term by multiplying both sides by and rearranging the terms.
Because the left-hand side of the equation is independent of phi and must be equal to the right-hand side for any value of phi, the equation must be equal to a constant which we will denote by . Thus, we now have an ordinary differential equation for phi and a partial differential equation involving theta and r.
“Azimulthal Equation”
“Theta-Radial Partial Differential Equation”
We continue or process of isolating variables on the remaining partial differential equation by dividing the equation by to isolate theta.
Since the left-hand side must equal the right-hand side of the equation regardless of the values of r and theta, the equation must equal a constant which we will denote by . This gives us our two final ordinary differential equations:
“Radial Differential Equation”
“Theta Equation”
E.Radial Equation
1. Orbital Angular Momentum and Effective Potential
In order to obtain some physical understanding about the radial equation and to put it into a form where we can find its solution in math handbooks, we are going to do some algebraic rearrangement to the equation.
We also note that the potential of our electron for the Coulomb potential is
Thus, we can rewrite the radial equation in a general form for any problem in which the potential depends only on the radial variable and not just the Coulomb potential as
.
By dimensional analysis and our past knowledge of the Schrodinger equation in one dimension, we know that the term in the bracket must represent a kinetic energy due to motion in the radial direction.
Thus, the last term in the bracket must be kinetic energy due to rotation (i.e. due to changes in theta and phi). Thus, we have the kinetic energy of rotation as
We know from PHYS122 that the kinetic energy of rotation is given as
where L is the magnitude of the object’s orbital angular momentum and I is the moment of inertia. For a point particle of mass , the moment of inertia is
.
Thus, our classical physics work gives us the rotational kinetic energy as
.
Comparing our classical physics results with the result from the radial part of the Schrodinger equation, we now observe relationship between the constant we used to separate the theta and radial equations and the maginitude of the orbital angula momentum.
“Orbital Angular Momentum Eigenvalues”
If we again look at the radial equation at the top of the page, we see that the rotational kinetic energy reduces the available energy for radial kinetic energy just as if it was a potential energy. This is a common occurrence in a wide range of physics problems involving radial forces including planetary motion. In these cases, physicists often chose to talk about an “effective potential” which is the sum of the actual potential due to forces and the kinetic energy due to rotation which is called the centrifugal barrier potential or just the ”centrifugal barrier.”
Thus, we can rewrite the radial equation in a more general form that is seen in many different fields of physics as
.
As an example of how our work transcends various problems, you should note that according to our formula the angular momentum of the moon can be thought of as a centrifugal barrier (repulsive potential) which prevents the moon from hitting the Earth. A similar argument explains why it is harder for a nucleus to undergo alpha decay between two nuclear states of different angular momentum. In alpha decay, the alpha particle must tunnel through the nuclear potential. In order to conserve angular momentum in decay, the difference in the angular momentum of two nuclear states must be offset by the angular momentum of the emitted alpha particle. Because the effective potential is raised when the alpha particle has angular momentum, the probability for tunneling is greatly reduced.
2.Solution for the Coulomb Potential
We now wish to change our approach from looking at the general nature of the radial equation to solving it for our specific case of the Coulomb potential.
a.Asymptotic Solution for Large Radial Distances
For large r, the differential equation is approximately,
This is an equation which we have solved several times previously. Since the wave function must stay finite at large values of r, the only acceptable solution is the exponential decay solution.
b.Ground State Wavefunction
Let us see if our asymptotic wavefunction is a possible solution to the hydrogen atom for all radial locations in some cases. We do this by using a trial wavefunction based upon our asymptotic results and plugging it into the full radial equation. Let us define the radial wave function in terms of a constant delta as
and find the necessary radial derivatives as
.
Plugging our results into the radial Schrodinger equation, we have
In order for the polynomial to be true for all values of r, the coefficients for each term must be zero!! This gives us three conditions that must be met for the trial function to be a solution for the hydrogen atom. The 2nd order coefficient gives us
The linear coefficient gives us
.
Solving for delta, we see that the constant is actually the first Bohr radius.
By combining results for the linear and 2nd order coefficients, we can determine the energy eigenvalue associated with our trial wavefunction.
Thus, we see that our trial wavefunction will be a solution of the hydrogen atom only if the energy is the ground state energy of the hydrogen atom.
We know look at the constant term.
This term will only be zero if or . We will see later that the theta equation restricts to non-negative values so we have found that . This implies that the electron in the ground state has zero angular momentum since in contradiction to the Bohr atom. The ground state electron actually has some intrinsic angular momentum. The Schrodinger equation doesn’t correctly predict this. Instead, we would need to use the Dirac equation from relativistic quantum mechanics.
with and
“Radial Wavefunction for Ground State of Hydrogen”
c.Asymptotic Solution for Small Radial Distances
Let us start by rewriting our radial differential equation as
.
For small values of r, we can simplify the differential equation as
.
One of the important things that we see when we write the equation in this form is that for small radius values the radial function is primarily effected by angular quantum number and not the wavefunctions energy eigenvalue E!!
Because the electron must have a zero probability of being found at r = 0, it is reasonable to choose a power solution for R as our test wavefunction of the form
where c is a constant.
Thus, we can now compute the required derivatives as
Plugging our results into the Schrodinger equation, we find that
In order for this to be true at all values of small r, the coefficient must be zero. Thus, we have shown that . This tells us that the radial wavefunction near the nucleus depends on the electron’s angular momentum!!
It is important that you realize how much information can be obtained without actually solving the full radial differential equation. This approach to problem solving is an important tool in your future work!!
d.Solving the Radial Schrodinger Equation in General
Eisberg and Resnick shows in Appendix N how to solve the general radial equation. I will leave the details for your reading, but I do want to point out some important facts that emerge from the process.
i)Each solution has a polynomial whose form depends on both the orbital angular quantum number and a second constant n which is called the “principle quantum number.”
ii)The constant n must be a non-zero positive integer and must be larger than the orbital angular quantum number in order for the radial equation to have a finite solution.
iii)The energy eigenvalue associated with each radial wavefunction depends only on principle quantum number and has the value .
The general form of the radial wavefunctions found by actually solving the radial Schrodinger equation can written in terms of the product of our asymptotic solutions and an unknown function.The general form of the radial wavefunctions in terms of the ratio of the radial distance to the Bohr radius is given by
where H(r/ao) is a polynomial whose form depends on two constants n and .
A few of the radial wavefunctions are provided below. For more functions look at Table 8.4 on page 280 of Serway/Moses/Moyer 3rd Edition as
F.Azimuthal (Phi) Equation
We now return to the azimuthal equation to obtain its solution.
We have seen this equation previously. In order for this equation to stay finite as we vary phi from minus infinity to positive infinity, it is necessary that (i.e. must be real). Thus, the solution is of the form of a complex exponential
.
Since we also require that the wave function be single valued at any point in space, the argument for the complex exponent must repeat anytime we increase or decrease phi by an integer multiple of two pi. Evaluating the function at phi equal zero gives us
We can determine the normalization constant by solving the equation
“Normalized Phi Wavefunction”
G.Theta Equation
We know return to the theta equation which can be written as
.
The solutions to this equation will depend on two quantum numbers, and .
To find the solution to this equation, we define a new variable, x, as.
We also find that . Using these results and the chain rule of Calculus, we can rewrite the theta equation as
.
We now use the fact that to change the equation to
This equation is known as Legendre’s Associated Differential Equation (see Eq. 28.49 page 165 of the Schaum’s Math handbook ). The solutions of this equation are well know special functions of engineering and physics (See Mathematical Methods for Physicists by Arfkin or similar textbooks). Eisberg and Resnick also go through the general solution technique in Appendix N.
The general results for the solution of Legendre’s Associated Differential Equation are the following:
i)The equation only has finite solutions if is a non-negative integer such that .
ii)The wavefunctions are normalized Associated Legendre Polynomials, , which are defined by the quantum number and .
H.Spherical Harmonics -
We can now combine our results for theta and phi to produce another famous set of special functions called spherical harmonics.
Spherical harmonics are tabulated in Table 8.3 on page 269 of Serway/Moses/Moyer 3rd Edition. You may also obtain them using relationship 28.50 for the Associated Legendre Polynomial on page 165 in the Schaum’s Math Handbook:
Example: Find
Solution:
First find
Now plug your result into the relationship for the spherical harmonics.
I.Z-Component of the Angular Momentum
We know from classical physics that orbital angular momentum is defined by the equation
.
From the definition, the x-component of the angular momentum is found to be
.
If we now use the operators for position and linear momentum that we discovered when developing the Schrodinger equation several modules ago, we have
.
However, we can use the chain rule of Calculus and the relationships between Cartesian coordinates and spherical coordinates (see Eisberg and Resnick Appendix M) to rewrite the z-component of the angular momentum’s operator as
Applying the operator to our spherical harmonics, we find that
.
Thus, we see that the spherical harmonics are eigenfunctions for the z-component of the angular momentum operator with eigenvalues of . Another interesting fact that comes out of this study is that there is an uncertainty relationship between the various components of the orbital angular momentum. You can know with perfect precision the magnitude of the angular momentum and the projection of the angular momentum upon one axis, but not its other two components!!
We summarize these results as follows:
i)The eigenfunctions for both angular momentum ands the z-component of angular momentum are the spherical harmonics.
ii)The z-component of the orbital angular momentum is determined by the magnetic quantum number, , by the relationship