Review of chapters 1 to 5

What have we learned so far?

Up to now, we were very careful to avoid using the “w” word as we wanted to avoid bringing a wave-function out of our hat (even if the hat is of quantum mechanical nature). It is a good time to take stock of what we have done so far. Essentially, our starting point was the Stern-Gerlach experiment we discussed in details and for which we introduced the notion of “state vectors” in Chapter 1: Stern-Gerlach Experiments. Next, we applied the rules of linear algebra to the space where those state vectors live and, naturally, we introduced operators (abstract objects that yield another state vector when applied to a state vector) in Chapter 2: Rotation of basis states and matrix mechanics. We then realized that observables (i.e., what we can measure) correspond to Hermitian operators. We spent an entire chapter (Chapter 3: Angular momentum) looking at the operation that consists in rotating the spin of a state vector and we introduced the very important tools related to infinitesimal rotations and their generator. This led to us realizing that because rotations do not commute (and, remember, rotations do not generally commute in the Cartesian space either!), the different components of the angular momentum operator do not commute. This allowed us to realize that the inability to know any two components of the angular momentum operators (as seen with the Stern-Gerlach experiment) was related to the fact that two operators that do not commute do not share a common set of eigenvectors. Next, in Chapter 4: Time evolution, we looked at another type of unitary transformation: time evolution. We quickly realized that the formal approach based on generators could be used to describe time evolution and this allowed us to introduce the energy operator, which is the Hamiltonian of the system (and thus the generator of time evolution). Finally, in Chapter 5: Combining two spin-1/2 particles, we graduated from single-particle systems to two-particle systems and realized that the combined state of the two-particle system could be pure or entangled. We spent quite a bit of time on the latter as they can lead to surprising results, including spooky action at a distance and quantum teleportation.

Test your knowledge

Consider the state vector $A(\ket{+z}+\frac{i}{\sqrt{2}}\ket{-z})$ to represent a spin-1/2 particle. Among the values of $A$ listed, which one makes this state a legal state?
1. 1
2. $\sqrt{\frac{2}{3}}e^{-i|\delta|}$
3. $\sqrt{\frac{2}{3}}e^{-|\delta|}$
4. $\frac{2}{3}$
5. Any value of $A$ is acceptable since the basis is complete.
Consider a spin-1/2 particle described by the state vector $\frac{1}{\sqrt{2}}(\ket{+z}+\ket{-z})$ entering a Stern-Gerlach SG:math:_z apparatus. The particle is deflected in the $+z$ direction on exit of SG:math:_z. What is the – legal – state vector describing the particle at the output?
1. $\frac{1}{\sqrt{2}}\ket{+z}$
2. $\ket{+z}$
3. $\frac{1}{\sqrt{2}}(\ket{+z}+\ket{-z})$
If a spin-1/2 particle is described by the state vector $\ket{\phi}=A(\ket{+x}+\frac{i}{3}\ket{-x})$ ; what is the expectation value of a measurement of $\hat{S}_x$ ?
1. 0
2. $\frac{4\hbar}{9}$
3. $\frac{4\hbar}{9}A^2$
4. $\frac{4\hbar}{9}|A|^2$
Imagine a particle with a total spin $s=0$ , can you know both the outcomes of SG:math:_x and SG:math:_y experiments with certainty?
1. No. The uncertainly principle always states that the product of uncertainties is a strictly positive number.
2. No. Quantum mechanics does not allow you to simultaneously know the outcome corresponding to two observables that do not commute.
3. Yes. I do not think we ever spoke about commutators in class. I’m pretty certain.
4. Yes. This is a trick question Professor. In general, you can’t! But for a particle-0, since there is only one possible outcome, there is no uncertainty.
If a ket vector is given by $\ket{\psi}=A\ket{+z}+B\ket{-z}$ , what is an equivalent representation of the corresponding bra vector among the possibilities below?
1. $\left(\begin{array}{l} A \\ B \end{array}\right)$
2. $\left(\begin{array}{l} A* \\ B* \end{array}\right)$
3. $\left(\begin{array}{ll} A & B \end{array}\right)$
4. $\left(\begin{array}{ll} A* & B* \end{array}\right)$
Calculate the expectation value and uncertainty for a measurement of ${\textrm{SG}_Z}$ for the following state: $\ket{+y}-\ket{-x}$ .
1. $-\hbar/2$ and 0
2. $-\hbar/2$ and $\hbar$
3. 0 and $\hbar$
An operator of rotation is always a unitary operator…
1. Yes, and its eigenvalues are all positive.
2. Yes, and its eigenvalues are all real.
3. Yes, and its eigenvalues are usually complex.
4. No, such an operator must be Hermitian.
5. There is not enough information to answer this question conclusively.
What are the eigenvalues of the generator of rotation for spin- $\frac{1}{2}$ along the $x$ -axis?
1. 0 and 1
2. $\pm \hbar/2$
3. $\pm \hbar$
4. ${\rm e}^{\pm i\phi/2}$
Calculate the following commutator: $\comm{\hat{J}_x}{\hat{J}_y+\hat{J}_z}$
1. $0$
2. $i\hbar(\hat{J}_z+\hat{J}_y)$
3. $i\hbar(\hat{J}_z-\hat{J}_y)$
4. $i\hbar(\hat{J}_x)$
It is possible to find a common set of vectors that simultaneously diagonalize the projections of the intrinsic spin angular momentum along $z$ ( $\hat{J}_z$ ) and the total spin operator ( $\hat{J}^2$ ).
1. True, since the two operators commute.
2. True, since there is no difference between classical and quantum physics.
3. False, since the two operators do not commute.
For a spin-3/2 particle, what is $\bra{\frac{3}{2},\frac{3}{2}}\hat{J}^2\ket{\frac{3}{2},\frac{3}{2}}$ ?
1. $\frac{15\hbar}{4}$
2. $\frac{15\hbar^2}{4}$
3. $\frac{3\hbar}{2}$
4. $\frac{3\hbar^2}{2}$
For a spin-5/2 particle, what is $\bra{\frac{5}{2},\frac{3}{2}}\hat{J}_{-}\ket{\frac{5}{2},\frac{3}{2}}$ ?
1. $-3\hbar/2$
2. $-\hbar/2$
3. $0$
4. $+\hbar/2$
5. $+3\hbar/2$
The time evolution operator preserves the norm of a state vector. Is this always true?
1. Yes
2. No
3. It depends on the type of Hamiltonian
Imagine the setup of a magnetic resonance experiment. There is a constant magnetic field along $z$ with $\omega_{0}=e g B_{0} / 2 m c$ and an oscillating magnetic field along $x$ with frequency $\omega$ and with $\omega_{1}=e g B_{1} / 2 m c$ . What is the most accurate answer from the claims below?
1. The resonance frequency of the system is $\omega=\omega_0$ . The energy absorbed and released by the spin of the system comes from (and is released to) an independent, separate source.
2. The resonance frequency of the system is $\omega=\omega_0$ . The energy absorbed and released by the spin of the system comes from (and is released to) the oscillating magnetic field.
3. The resonance frequency of the system is $\omega=\omega_1$ . The energy absorbed and released by the spin of the system comes from (and is released to) an independent, separate source.
4. The resonance frequency of the system is $\omega=\omega_1$ . The energy absorbed and released by the spin of the system comes from (and is released to) the oscillating magnetic field.
In Quantum Mechanics, how do you define a stationary state?
1. This is a state whose spatial location is constant.
2. This is a state whose energy does not change during the time evolution.
3. This is a state for which the expectation value of the linear momentum operator is exactly zero.
An operator $\hat{O}$ commutes with the Hamiltonian of a system at $t=0$ . What claim below is the most correct?
1. The expectation value $\expval{O}$ is constant.
2. The expectation value $\expval{O}$ is constant, provided that $\hat{O}$ is not time-dependent.
3. We can’t say much about the time evolution of the expectation without more information about the system.
The EPR paradox demonstrated that it is necessary to have hidden variables in order to make sense of quantum mechanics.
1. False. It turns out there is no paradox, as demonstrated by Bell’s theorem and subsequent experimental evidence.
2. True. To this day, the theory of hidden variables is the accepted interpretation of quantum mechanics.
3. Well, it depends what EPR means. It may mean Enjoy Physics at Rensselaer but I’m not so sure.
Imagine 2 particles of spin-1. What is the dimension of the corresponding space where their combined state vectors live?
1. 1
2. 3
3. 6
4. 9
In the hydrogen atom, the hyperfine Hamiltonian represents:
1. Something that is of extremely high-quality.
2. The interaction between the magnetic moment of the proton and that of the electron.
3. The electrostatic interaction between the proton and the electron due to their electrical charges
Consider a system of two particles of spin-1/2. Also consider these two basis sets:

Basis 1:

$\left\{\begin{array}{ll} \ket{1}=\ket{+z}_1\ket{+z}_2;& \ket{2}=\ket{+z}_1\ket{-z}_2;\\ \ket{3}=\ket{-z}_1\ket{+z}_2;& \ket{4}=\ket{-z}_1\ket{-z}_2 \end{array}\right.$

Basis 2:

$\left\{\begin{array}{ll} \ket{1}=\frac{1}{\sqrt{2}}(\ket{+z}_1\ket{-z}_2+\ket{-z}_1\ket{+z}_2);& \ket{2}=\frac{1}{\sqrt{2}}(\ket{+z}_1\ket{-z}_2-\ket{-z}_1\ket{+z}_2);\\ \ket{3}=\frac{1}{\sqrt{2}}(\ket{+z}_1\ket{+z}_2+\ket{-z}_1\ket{-z}_2);& \ket{4}=\frac{1}{\sqrt{2}}(\ket{+z}_1\ket{+z}_2-\ket{-z}_1\ket{-z}_2) \end{array}\right.$

We could use either basis to describe the system of two particles with spin 1/2. Both of them are actually a set of eigenvectors of the total angular momentum of the system.
We could use either basis to describe the system of two particles with spin 1/2. However, neither of them is a complete set of eigenvectors of the total angular momentum of the system.
Only the first basis should be used since we can’t use a basis of entangled states in quantum mechanics.
The second basis rings a bell but I am not sure about the answer.

Hint

Find the answer keys on this page: Answers to selected test your knowledge questions. Don’t cheat! Try solving the problems on your own first!