Exercises 5.7 Problems
1.
Write either a poem, a song, or a few sentences explaining what it is about spin — on a quantum level — that can't be explained by classical laws of physics.
2.
Suppose an electron is known to have a \(z\)-component of spin \(S_z\) of \(+\hbar/2\text{;}\) that is, we know it is in the \(| +z \rangle\) state.
Find the probability that a measurement of its spin along the \(x\)-axis gives the value \(+\hbar/2\text{.}\)
Find the probability that a measurement of its spin along the \(y\)-axis gives the value \(-\hbar/2\text{.}\)
3.
Assume that a particle is in the state
where \(\vert \phi_1 \rangle\) and \(\vert \phi_2 \rangle\) are a normalized set of basis states.
A measurement is made on the particle. Calculate the probability that this measurement will result in a value corresponding to the state \(\vert \phi_1 \rangle\text{.}\)
Let's say that the measurement made in part (a) does, in fact, produce a result corresponding to the state \(\vert \phi_1 \rangle\text{.}\) The measurement is then immediately repeated. Calculate the probability that this second measurement will result in a value corresponding to the state \(\vert \phi_2 \rangle\text{.}\)
Another particle is prepared in the original state \(\vert \psi \rangle\text{.}\) A measurement is made on this new particle. Calculate the probability that this measurement will result in a value corresponding to the state \(\vert \phi_2 \rangle\text{.}\)
4.
An electron is known to be in the spin state \(|\psi\rangle = \frac{3}{5}| +z \rangle + \frac{4}{5}| -z \rangle\text{.}\)
The electron is sent through a device that measures its spin angular momentum along the \(x\)-direction. Compute the probability of obtaining the result \(+\hbar/2\) for the \(x\)-component of spin. Compute the probability of obtaining \(-\hbar/2\) for this measurement. (Check to make sure that your probabilities add up to 1.)
The electron initially in the state \(|\psi\rangle\) specified above is sent through a device that measures its spin angular momentum along the \(y\)-direction. Compute the probability of obtaining the result \(+\hbar/2\) for the \(y\)-component of spin. Compute the probability of obtaining \(-\hbar/2\) for this measurement.
Note that you get different answers for parts (a) and (b). Mathematically, this comes from the factor of “\(i\)” in the equations for \(| +y \rangle\) and \(| -y \rangle\text{.}\) Conceptually, this is another example of some quantum weirdness — you would think that a state that is a superposition of stuff solely in the \(z\)-direction would give the same results for \(x-\) and \(y\)-components, but that isn't the case. Think about this for a moment, and convince yourself that this is weird.
5.
Assume that you have an electron that is in a superposition of “spin-up” and “spin-down” states:
You measure the vertical component of spin. What is the probability that you will find \(S_z = + \hbar/2\text{?}\)
Another electron is in the same state \(|\psi_1\rangle\text{.}\) You now measure the \(y\)-component of spin. What is the probability that you will find \(S_y = + \hbar/2\text{?}\)
-
Another electron is prepared in a superposition state
\begin{equation*} |\psi_2\rangle = \sqrt{\frac{1}{5}}\vert +z \rangle + \sqrt{\frac{4}{5}} | -z \rangle\text{.} \end{equation*}What is the probability that a measurement of this electron's \(y\)-component of spin will find \(S_y = + \hbar/2\text{?}\)
Now, assume that you do find \(S_y = + \hbar/2\text{.}\) You measure \(S_y\) again for the same particle. What is the probability that you'll find \(S_y = -\hbar/2\text{?}\)
6.
A particle is in the state given by
where \(\vert \psi_1 \rangle\) and \(\vert \psi_2 \rangle\) are a normalized set of basis states.
A measurement is made on the particle. Calculate the probability that the measurement will result in a value corresponding to the state \(\vert \psi_1 \rangle\text{.}\)
Another particle is prepared in the state \(\vert \phi \rangle\text{,}\) and a measurement is made on this particle. Calculate the probability that this measurement will result in a value corresponding to the state \(\vert \psi_2 \rangle\text{.}\)
7.
The quantum state of a photon propagating along the \(z\)-direction can be written in terms of the states \(\vert X \rangle\) and \(\vert Y \rangle\text{,}\) where \(\vert X \rangle\) represents a photon polarized along the \(x\)-direction and \(\vert Y \rangle\) represents a photon polarized along the \(y\)-direction. A photon in an arbitrary polarization state can be written as a linear superposition of the states \(\vert X \rangle\) and \(\vert Y \rangle\) as
where \(\theta\) is the angle of polarization of the photon measured with respect to the \(x\)-axis.
Assuming that the states \(\vert X \rangle\) and \(\vert Y \rangle\) form a basis, show that the state \(\vert\theta\rangle\) is normalized.
A polarizer is a device that measures the polarization of a photon along a certain direction. Let's say that a polarizer is oriented along the \(x\)-direction, that is, it is measuring for the state \(\vert X \rangle\text{.}\) For an incident beam of photons in the state \(\vert \theta \rangle\) as given above, what is the probability that the polarizer measures the state \(\vert X \rangle\) (i.e., the photons go through the polarizer)? What is the state of the photons after making this measurement?
Let's say we have a beam of photons in polarization state \(\vert X \rangle\) incident on a polarizer oriented in a direction making an angle \(\theta\) with respect to the \(x\)-axis. What is the probability that the photons pass through the polarizer oriented at angle \(\theta\text{?}\) (Hint: think about the previous question in the reverse order of events). What is the state of the photons after making this measurement?
Following up on part (c), the beam in state \(\vert \theta \rangle\) is incident on a second polarizer oriented along the \(y\)-axis. What is the probability that the photons pass through the second polarizer?
Now, putting this all together, photons pass through polarizer #1 oriented along the \(x\)-axis, then pass through polarizer #2 oriented at an angle \(\theta\text{,}\) then on to polarizer #3 oriented along the \(y\)-axis. What is the total probability that a photon passing through polarizer #1 will also pass through polarizer #3? [Note: Compare your result with what you did in Lab #18 “Polarization of Light.” ]
8.
Using a technique similar to that used in Example 5.8, show that you can write the spin-down \(S_z\) state \(\vert -z \rangle\) as a linear superposition of the states \(\vert +x \rangle\) and \(\vert -x \rangle\) Compare your results with those given in Table 5.9.
9.
Using equations (5.18) and (5.19), write expressions for the states \(\vert +z \rangle\) and \(\vert -z \rangle\) in terms of linear combinations of the states \(\vert +y \rangle\) and \(\vert -y \rangle\text{.}\) Compare your results with those given in Table 5.9.
10.
An electron is placed in a spin state given by
Calculate the probability of obtaining a value of \(-\hbar/2\) when the \(z\)-component of spin \(S_z\) is measured.
Calculate the probability that an electron in state \(\vert \psi \rangle\) will be measured to have an \(x\)-component of spin \(S_x\) of \(+\hbar/2\text{.}\)
11.
An electron is placed in the spin state
as is the case in Example 5.10. An experiment is performed to measure the \(y\)-component \(S_y\) of spin. Calculate the probability that this measurement results in a value of \(-\hbar/2\text{.}\)
12.
An electron in a particular system can have any of the discrete energies \(E_n = (-8.00\Xunits{eV})/n^2\text{,}\) where \(n = 1, 2, 3, \dots\text{.}\) Assume that the electron is in a state
Show that this state is normalized.
If the energy of the electron is measured, what is the probability that the result of the measurement will be \(E_2 = -2.0\Xunits{eV}\text{?}\) What is the probability of the measured energy being \(E_4 = -0.5\Xunits{eV}\text{?}\)
What is the probability that the result of measuring the energy would be either \(E_1\text{,}\) \(E_3\text{,}\) or \(E_5\text{?}\)
Assume that 10,000 electrons are prepared to be in the same state \(|\psi\rangle\) and that the energy is measured for each of these electrons. Approximately how many of these electrons will be measured to have energy \(E_1\text{?}\) \(E_2\text{?}\) \(E_3\text{?}\) \(E_4\text{?}\) \(E_5\text{?}\)
13.
Protons placed in a magnetic field can be either in the spin-up \(\vert +z \rangle\) or spin-down \(\vert -z \rangle\) state with an energy difference between these two states \(\Delta E_\text{ proton }\text{.}\) Incident photons of frequency \(2.20 \Xunits{MHz}\) are absorbed causing transitions of the protons from the spin-up \(\vert +z \rangle\) state to the spin-down \(\vert -z \rangle\) state. Determine the magnitude of the total magnetic field \(B\) in which these protons are placed.
14.
The bulk of gas in our galaxy is atomic hydrogen (composed of a proton and an electron) in the ground electronic state. Astronomers map the location of this neutral hydrogen by detecting photons that are emitted when the spin of the electron flips from being parallel with the spin of the proton to being anti-parallel. The effective magnetic field experienced by the electron in a hydrogen atom is \(0.0507 \Xunits{T}\text{.}\) Determine the wavelength of the photon emitted when the electron undergoes a spin flip.
15.
In the presence of a strong magnetic field, the energy levels of an atom change due to the interaction of the electronic spins with the magnetic field. This is known as the “anomalous” Zeeman effect, and can be used to determine the strength of magnetic fields on the Sun.
Magnetically active regions of the Sun have typical magnetic fields strengths of \(0.4 \Xunits{T}\text{.}\) Determine the energy difference (in eV) between electrons that are aligned with the magnetic field and anti-aligned with the magnetic field.
This difference in energy causes a split in each electronic energy level of the atom. Imagine that we have a 2-level atom where the 1\(^\text{ st }\) excited state is \(0.1 \Xunits{eV}\) above the ground state. Draw energy level diagrams for this atom: i) when there is no magnetic field and ii) in the presence of the magnetic field from part (a).
How would we expect the spectrum of photons absorbed by this atom to change in the presence of a strong magnetic field versus in the absence of a magnetic field? How would this spectrum change as the strength of the magnetic field increases?
16.
Let's use a simulation of the Stern-Gerlach experiment to explore quantum states and spin. Go to
http://phet.colorado.edu/sims/stern-gerlach/stern-gerlach_en.html By default, the atoms start in the state \(\vert +x \rangle\text{,}\) and the device is setup to make a measurement of the \(z\)-component of spin.
To get a feel for how this works, try firing a few atoms. You'll notice a counter at the bottom of the page keeping track of how many atoms have been measured to be spin up or spin down as measured along the \(z\) axis. Can you predict whether an individual atom will have a measured \(z\)-component of spin up or down?
For atoms starting in the state \(\vert +x \rangle\text{,}\) what is the probability that a measurement of the \(z\)-component of spin results in \(S_z = +\hbar/2\text{?}\) To determine this, turn on “Auto Fire” and keep it running until the percentages are not longer changing.
For atoms starting in the state \(\vert +x \rangle\text{,}\) what is the probability that a measurement of the \(x\)-component of spin results in \(S_x = +\hbar/2\text{?}\) To measure the \(x\)-component of spin, change the “angle” in the simulation to \(-90^\circ\text{.}\)
Now, change the number of magnets to 2. Orient the first magnet so that it is measuring the z-component of spin (angle = 0) and orient the second magnet so that it is measuring the x-component of spin (\(\mbox{angle} = -90^\circ\)). In this setup the simulation starts with atoms in the state \(\vert +x \rangle\) and measures the \(z\)-component of spin. For atoms with measured \(S_z = +\hbar/2\text{,}\) it then measures the \(x\)-component of spin. For an atom that started in the state \(\vert +x \rangle\text{,}\) for which we then measure \(S_z = +\hbar/2\text{,}\) what is the probability that a subsequent measurement of the \(x\)-component of spin results in \(S_x = +\hbar/2\text{?}\)
Based on your answer to part (c), what is the state of the atom immediately after a measurement of its \(z\)-component of spin results in \(S_z = +\hbar/2\text{?}\) Hint: Think about collapse of state!
17.
Suppose the possible energies that a particle can have are \(E_1\text{,}\) \(E_2\text{,}\) \(E_3\text{,}\) \(E_4\text{,}\) … Consider a particle in a state given by the following linear superposition of energy basis states:
where \(c_4\) is an undetermined constant.
Calculate the probability of obtaining a value \(E_3\) in a measurement of the energy of this particle.
The particle is prepared again in the state \(|\psi\rangle\text{.}\) Calculate the probability of obtaining a value \(E_2\) in a measurement of the energy of this particle?
How many possible values are there for the constant \(c_4\text{?}\) Determine two possible values for this constant.
18.
A quantum particle is in the following superposition of energy states
Determine the probability that a measurement of the energy will yield the value \(E_3\text{.}\)
You measure the energy and do, in fact, find the energy to be \(E_3\text{.}\) Write down a correct expression for the new state of the particle just after this energy measurement.
19.
Each of the following diagrams shows a beam of electrons inititally with random spin orientations passing through a series of Stern-Gerlach (SG) apparatuses of different orientations. In each case, determine whether a beam will come out of one, both, or neither of the two exits of the final SG apparatus, and also determine how the intensity of any non-zero beam will compare with the intensity of the initial beam of electrons.
20.
Consider the following Stern-Gerlach (SG) experiment. A beam of electrons, prepared in a certain state \(\ket{\psi}\text{,}\) is emitted from an electron source. The beam passes into a Stern-Gerlach device which measures the \(z\)-component of spin \(S_z\) as shown below. A total of 1000 electrons go through this apparatus, of which exactly 400 emerge with \(S_z=+\hbar/2\) (spin-up along the \(z\)-axis) and exactly 600 emerge with \(S_z=-\hbar/2\) (spin-down along the \(z\)-axis).
Write a possible normalized state \(\ket{\psi}\) for the electrons in the initial beam that corresponds to the observations in this experiment.