Exercises 6.10 Problems
1.
In filling up the energy states of an infinite square well potential while obeying the Pauli exclusion principle, it is often stated that one electron can go in spin up and the other electron can go in spin down. This statement is reasonably correct, but it becomes incorrect if you interpret it to mean that the two electrons are in the state \(|E_1\hspace{-1mm}\uparrow \hspace{2mm} E_1\hspace{-1mm}\downarrow\rangle\)
Explain why this cannot describe the two electrons in the ground state.
Write down the correct state that is intended when two electrons are put into the ground state.
2.
Consider the case of an electron and a muon (the electron's cousin, also a spin-1/2 fermion) in a two-particle quantum state.
Is \(|e\hspace{-1mm}\uparrow\hspace{2mm} \mu\hspace{-1mm}\downarrow\rangle\) a possible two-particle state? Explain your reasoning.
Is \(|e\hspace{-1mm}\uparrow\hspace{2mm} \mu\hspace{-1mm}\uparrow\rangle\) a possible two-particle state? Explain your reasoning.
3.
Consider two identical particles that are to be put into two single-particle states labeled by \(|\alpha\rangle\) and \(|\beta\rangle\text{.}\)
Make three charts like Figure 6.2 describing all the possible ways that you can make a two-particle state for classical particles, bosons, and fermions. Use the symbols \(\bullet\) and \(\circ\) to represent the two distinguishable classical particles, and two \(\bullet\)'s to represent the indistinguishable quantum particles.
How many states are there for the classical particles? For the bosons? For the fermions?
Assuming that the microstates are equally likely, what is the probability of finding the two particles in the same single-particle state for classical particles? For bosons?
Are your results consistent with the idea that bosons have an enhanced probability (relative to classical particles) of being in the same state?
4.
Neutrons have spin \(s=1/2\text{.}\) Six neutrons are placed in an infinite square well with ground state energy \(E_1 = 3\Xunits{eV}\text{.}\) Determine the minimum combined energy these neutrons can have.
5.
Explain briefly why a population inversion is necessary for the operation of a laser.
6.
The sketches in Figure 6.7 show the state of a two-level atom and possibly a photon. For each “Before” sketch, make a corresponding “After” sketch and name the process.
7.
Superconductors. (Do in Problem Session) Here we investigate some magnetic properties of superconductors.
Closely observe the little cube hovering over the disk. Comment on what you observe. What evidence do you have that this is a superconductor? Can you make the cube spin?
Explain how the superconductor can levitate the magnet.
8.
Why would you not expect helium-3 to act as a superfluid at low temperatures, even though helium-4 does?
9.
Sketch the magnetic field lines outside the north-pole end of a bar magnet (the field points generally away from the north pole). Now, using the right-hand rule that relates magnetic field to current, sketch the shielding currents that must flow in a flat superconducting plate when the north-pole end of the bar magnet is held just above the plate. Pay particular attention to the direction of the shielding current. Does the current in the plate repel or attract the bar magnet?
10.
Superconducting magnets. One of the most productive uses of superconductors is in the fabrication of strong electromagnets (e.g., for magnetic resonance imaging in hospitals, or in massive particle accelerators). Consider a superconducting magnet constructed from a solenoid of 1000 turns of superconducting wire with radius \(50\, \mbox{cm}\) and length \(1\, \mbox{m}\text{.}\) A current \(I\) is applied, resulting in a magnetic field of 10 Tesla. Once the magnetic field is produced, what power would have to be provided to the electromagnet to maintain the magnetic field?
11.
What would happen if dice were indistinguishable bosons? Of course, real dice are too large for quantum effects to be significant, so they are classical, distinguishable objects. As a result, rolling a pair of classical dice yields 36 possible results (1-1, 1-2, 1-3, …, 6-5, 6-6).
Calculate the probability of rolling doubles with real (classical) dice.
If dice were indistinguishable bosons, then rolling a 2-5 combination would be exactly the same as rolling a 5-2 combination. How many different results are possible for bosonic dice?
How many different ways of rolling doubles are possible with bosonic dice?
Use your results from (a) and (b) to calculate the probability of rolling doubles with bosonic dice. Is the probability higher, lower, or the same as the classical result?
12.
Is it possible to create a quantum state in which one electron in an infinite square-well potential is in the \(n=2\) state with spin up and another electron is in the ground state (\(n=1\)) with spin up? If so, write it out and make sure that it is antisymmetric. If not, explain why not.
13.
Consider three identical particles that are to be put into two single-particle states labeled by \(|\alpha\rangle\) and \(|\beta\rangle\text{.}\)
Make three charts like Figure 6.2 describing all the possible ways that you can make a three-particle state for classical particles, bosons, and fermions. Use the symbols \(\bullet\text{,}\) \(\circ\) and × to represent the three distinguishable classical particles, and three \(\bullet\)'s to represent the indistinguishable quantum particles.
How many states are there for the classical particles? For the bosons? For the fermions?
Assuming that the microstates are equally likely, what is the probability of finding the three particles all in the same single-particle state for classical particles? For bosons?
Show how your results are consistent with the idea that bosons have an enhanced probability (relative to classical particles) of being in the same state, and consistent with the Pauli exclusion principle.
14.
A ruby laser is a solid-state laser that uses a synthetic ruby crystal as the medium in which stimulated emission occurs as opposed to individual atoms in a gas laser. The diagram below shows the energy levels used in the ruby crystal along with their approximate energies relative to the ground state.
What is the frequency of the radiation needed to optically pump the ruby crystal and therefore create the necessary population inversion?
What is the wavelength of the emitted laser beam due to stimulated emission?
15.
A superconducting wire of circular cross section has a radius of \(1.5\Xunits{cm}\text{.}\) The critical magnetic field that this particular superconductor can withstand is \(B_c=20\Xunits{T}\text{.}\) Based on this information, calculate the maximum current that this superconducting wire is capable of carrying. Hint: use Ampere's law.
16.
This problem refers to Figure 6.2, which shows all the possible ways to put two particles into three states \(\ket{\alpha}\text{,}\) \(\ket{\beta}\text{,}\) and \(\ket{\gamma}\text{.}\)
For classical particles, take the solid circle to represent particle 1 and the open circle to represent particle 2. Then the two-particle state represented in row 4 would be written \(\ket{\alpha\;\beta}\text{,}\) while the two-particle state in row 7 would be \(\ket{\beta\; \alpha}\text{.}\) Using this notation, write out the classical two-particle states for each of the nine states given.
For bosons, the two-particle states must be symmetric under interchange of particles 1 and 2. Write out the appropriate two-particle states for each of the six states given.
For fermions, the two-particle states must be antisymmetric under interchange of particles 1 and 2. Write out the appropriate two-particle states for each of the three states given.