Quantum Rules and Stern-Gerlach
Particles whose x-spin state is measured to be |+x> are sent through an SGz box, and the particles that are measured to be |+z> are sent into an SGx box. What fraction of the original |+x> particles will now be found to be |-x>?
25%
What is a product state?
simplest way to describe a system made of two quantum particles; represents a scenario where particles are independent of each other.
In what basis must a state be expressed in order to apply the time evolution rule?
Energy eigenstate basis
What does it mean to 'normalize' the wavefunction?
Multiply psi by an overall value such that the total probability of getting any outcome is 1.
Particles whose x-spin state is measured to be |+x> are sent through an SGz box, and the particles that emerge from both box outputs are sent, without measuring their spin, into an SGx box. What fraction of the original |+x> particles will now be found to be |-x>?
0%
What are the two key features of an entangled state?
The two particles do not have definite values of an observable (such as z-spin, or photon polarization), but there is a definite correlation between the measurement outcomes (i.e. both have the same spin or both have different spin)
What is meant by the 'expectation value' of a measured quantity?
The average value that would be obtained after a very large number of measurements on identical quantons
In the photoelectric effect experiment, which of the following properties of the incoming photons must equal or exceed a minimum value in order for electrons to be ejected (choose all that are correct)? Choose from: wavelength, frequency, energy, number of photons
frequency and energy
If |a> and |b> are two quantum states, what is the physical meaning of the inner product <a|b>?
<a|b> is the probability amplitude that if the quanton starts in state |b>, it will be measured to be in state |a>. (This is absolute squared to get the probability of that measurement.)
What is the significance of Bell's theorem?
Provided a way in which QM and hidden variables would make different predictions, and therefore could be tested
Which of the following quantities do NOT change as a quantum state undergoes time evolution? Choose all correct answers from the following options: Probability of measuring one of the energy eigenstates, expectation value of energy, probability of measuring a state that is not an energy eigenstate, expectation value of a variable whose eigenstates are not energy eigenstates)
Probability of measuring one of the energy eigenstates, expectation value of energy
When placing 6 electrons in the lowest-energy arrangement in a 1D box, considering the Pauli Exclusion Principle, how many energy levels have electrons in them?
Three — two electrons with opposite spins can go in each level, so that each electron has a unique quantum state
What does it mean for two observables to be 'incompatible'?
This means that the eigenstates of one observable are superpositions of the other — a quanton cannot have definite values of both observables at once.
Give an example of an entangled state of two spins (label them A and B) that are positively correlated (always the same) and the collapsed state that results after measurement of the z-spin of A
|psi> = [|+z>A|+z>B + |-z>A|-z>B]/sqrt(2)
collapsed state = |+z>A|+z>B
or could be |-z>A|-z>B
Why are neutrino flavors not stable over time?
Because they are superpositions of energy eigenstates, not energy eigenstates themselves, and over time the relative phase of the superposition changes, changing its identity
Suppose an electron is in the n = 3 level of a 1D box. What are the possible wavelengths of photons that could be emitted from this system, in terms of the energy of the n = 1 level, E1 (and any needed constants)?
The energies of these transitions are 5E1 and 8E1, corresponding to wavelengths of hc/5E1 and hc/8E1