A particle of mass is free to move along the x-direction such that . The state of the system is represented by the wavefunction Eq. (4.74)
with given by Eq. (4.59).
Note that I’ve inserted a factor above that isn’t in the text, because otherwise will not be unit normalized (assuming is normalized in wavenumber space).
(a) What is the group velocity associated with this state?
(b) What is the probability for measuring the particle at position at time ?
(c) What is the probability per unit length for measuring the particle at position at time ?
(d) Explain the physical meaning of the above results.
(a). group velocity.
To calculate the group velocity we need to know the dependence of on .
Let’s step back and consider the time evolution action on . For the free particle case we have
Writing we have
Each successive application of will introduce another power of , so once we sum all the terms of the exponential series we have
Comparing with 1.1 we find
This completes this section of the problem since we are now able to calculate the group velocity
(b). What is the probability for measuring the particle at position at time ?
In order to evaluate the probability, it looks desirable to evaluate the wave function integral 1.4.
Writing , the exponent of that integral is
The portion of the exponential
then comes out of the integral. We can also make a change of variables to evaluate the remainder of the Gaussian and are left with
Observe that from 1.2 we can compute , which could be substituted back into 1.7 if desired.
Our probability density is
With a final regrouping of terms, this is
As a sanity check we observe that this integrates to unity for all as desired. The probability that we find the particle at position is then
The only simplification we can make is to rewrite this in terms of the complementary error function
Sanity checking this result, we note that since the probability for finding the particle in the range is as expected.
(c). What is the probability per unit length for measuring the particle at position at time ?
This unit length probability is thus
(d). Explain the physical meaning of the above results.
To get an idea what the group velocity means, observe that we can write our wavefunction 1.1 as
We see that the phase coefficient of the Gaussian “moves” at the rate of the group velocity . Also recall that in the text it is noted that the time dependent term 1.11 can be expressed in terms of position and momentum uncertainties , and . That is
This makes it evident that the probability density flattens and spreads over time with the rate equal to the uncertainty of the group velocity (since ). It is interesting that something as simple as this phase change results in a physically measurable phenomena. We see that a direct result of this linear with time phase change, we are less able to find the particle localized around it’s original time position as more time elapses.
A particle with intrinsic angular momentum or spin is prepared in the spin-up with respect to the z-direction state . Determine
and explain what these relations say about the system.
Solution: Uncertainty of with respect to
Noting that we have
The average outcome for many measurements of the physical quantity associated with the operator when the system has been prepared in the state is .
We could also compute this from the matrix representations, but it is slightly more work.
Operating once more with on the zero ket vector still gives us zero, so we have zero in the root for 2.16
What does 2.20 say about the state of the system? Given many measurements of the physical quantity associated with the operator , where the initial state of the system is always , then the average of the measurements of the physical quantity associated with is zero. We can think of the operator as a representation of the observable, “how different is the measured result from the average ”.
So, given a system prepared in state , and performance of repeated measurements capable of only examining spin-up, we find that the system is never any different than its initial spin-up state. We have no uncertainty that we will measure any difference from spin-up on average, when the system is prepared in the spin-up state.
Solution: Uncertainty of with respect to
For this second part of the problem, we note that we can write
So the expectation value of with respect to this state is
After repeated preparation of the system in state , the average measurement of the physical quantity associated with operator is zero. In terms of the eigenstates for that operator and we have equal probability of measuring either given this particular initial system state.
For the variance calculation, this reduces our problem to the calculation of , which is
so for 2.22 we have
The average of the absolute magnitude of the physical quantity associated with operator is found to be when repeated measurements are performed given a system initially prepared in state . We saw that the average value for the measurement of that physical quantity itself was zero, showing that we have equal probabilities of measuring either for this experiment. A measurement that would show the system was in the x-direction spin-up or spin-down states would find that these states are equi-probable.
I lost one mark on the group velocity response. Instead of 3.23 he wanted
since peaks at .
I’ll have to go back and think about that a bit, because I’m unsure of the last bits of the reasoning there.
I also lost 0.5 and 0.25 (twice) because I didn’t explicitly state that the probability that the particle is at , a specific single point, is zero. I thought that was obvious and didn’t have to be stated, but it appears expressing this explicitly is what he was looking for.
Curiously, one thing that I didn’t loose marks on was, the wrong answer for the probability per unit length. What he was actually asking for was the following
That’s a whole lot more sensible seeming quantity to calculate than what I did, but I don’t think that I can be faulted too much since the phrase was never used in the text nor in the lectures.