PHYS 221: Practice Exam Problems – Part 2 Spring 2022
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Physics 221 Spring 2022 – Document #16: Practice Exam Problems – Part 1
PHYS 221: Practice Exam Problems – Part 2
2022
Problem 3. A 3-D Isotropic Oscillator
The Schrodinger Equations for a 1-D Quantum Harmonic Oscillator can be written in this way:
∇2 ψ(~r) + U(~r)ψ(~r) = Eψ(~r)
where
U(~r) = 1
2mω 2r2
In contrast to the hydrogen atom (Coulomb potential) for the 3-D oscillator we can easily restate the potential energy in terms of a simple function of Cartesian coordinates:
U(~r) = 1
2mω 2r2 = 1
2mω 2 (x2 + y2 + z2 ) = U(x,y,z)
By using Separation of Variables it can be shown that the energy levels are given by three quantum numbers:
Enx ny nz = 
nx + ny + nz + 3
2
ω
a. By analogy with the 1-D SHO, what is the ground-state energy of the 3-D oscillator?
b. Suppose you measure the energy of the oscillator to be 9
2
ω . What is the degeneracy of this energy level? In other words, how many different individual states would give this energy?
c. In the equation above, we assume the oscillator is“isotropic”which means that ω = p
m means that the spring constant κ is the same in all directions. Suppose however, that we change
the problem so that κx is slightly larger than κy = κz . How will this split the energy levels listed in Part (b)?
Problem 4. Spherical Coordinates
a. A point is located in space with Cartesian coordinates (x,y,z) = (1, 2, 3). Express the position of this point in spherical coordinates: (r,θ,φ)
b. A meteor is a rocky object that enters the earths atmosphere and travels in a straight line. A student of astronomy sees what appears to be a meteor located at a heading (r,θ,φ). The object is observed to travel so that r and θ remain constant. The student says“this object is maintaining a constant height.”Is the student correct? Explain. Is the object a meteor? Explain.
Problem 5. A Dilithium Molecule
There is a real substance called “dilithium” comprising two lithium atoms (Z=3) covalently bonded together, generally found in the gas phase. We can consider the dilithium molecule be a “quantum system” corresponding to a single wave function to describe the two atoms bound to- gether due to interactions of one valence electron from each atom. The spatial wave function for the two electrons is known to be symmetric (unchanged) with regards to electron particle interchange.
a. Given that each atom is identical, and that each atom contributes one valence electronic to combined wave function that forms the molecular bond, describe the spin state for the combined two electrons. Use “up” and “down” arrow notation. Is this spin state symmetric or antisymmetric? How do you know?
b. In principle, two electrons can be combined to form a combined wave function that is spatially antisymmetric. In this case, indicate the possible spin states for the two electrons, again using “up” and “down” arrow notation.
c. In practice, diatomic molecules that are bound together are only found with spatially sym- metric wave functions. Such wave functions are called “bonding” while the corresponding spa- tially antisymmetric wave functions are called “anti-bonding” . Explain in qualitative terms while a spatially symmetric wave function works to form the molecule while the spatially antisymmetric wave function does not. Think about this in terms of where you are likely to find the two electrons relative to each other.
Problem 6. Carbon atom
Consider the carbon atom with atomic number = 6.
a. Write down the “orbital notation” for carbon in the ground state.
b. Write down the possible values of ℓ corresponding to the total orbital angular momentum of the two valence electrons. Hint: there are three of these.
c. Write down the possible values of s corresponding to the total spin angular momentum of the valence electrons. Hint: there are two of these.
d. In principle, what are all of the possible values of the total angular momentum quantum number j for this atom?
e. In fact, not all possible combinations of s and ℓ are allowed here because of the exclusion principle. For example, the state where j = 3 is not allowed. Can you explain why this is so?
2022-04-09