19PHB104 Quantum Mechanics

Learning outcomes
This coursework assesses your ability to
1. analyse the results of a computer simulation of an experiment.
2. solve problems by using the appropriate mathematical techniques.
3. describe quantum states of discrete systems and discuss the interpretation of quantum mechanics.

 

Important information
1. This assignment constitutes 20% of the module assessment.

 

2. All questions should be answered. The coursework is marked out of 100 for overall achievement of the learning outcomes according to the marking rubric shown.

 

3. The coursework must be submitted through Learn by the deadline specified.

 

4. The coursework should be typeset or word processed and submitted if possible as a PDF, as other formats can occasionally become corrupted.

 

5. The coursework will be marked within three weeks of the submission deadline and discussed at the next available teaching session.

 

6. Late submission without an authorised extension will receive a mark of zero. Extensions of up to 48 hours can only be granted under exceptional circumstances and according to the procedure expect to have any problems please get in touch as soon as possible. If you are unable to complete the work on time, you are advised to submit as much as you can by the deadline.

 

7. Where you draw on the work of others, you must cite the source in the appropriate format.
Failure to do this could result in a charge of academic misconduct.

 

8. Questions regarding the content of the coursework must only be raised during teaching sessions or through the Quantum Mechanics discussion forum on Learn.

 

9. Marking is anonymous. Please include your ID number, but not your name, in your submission and in the filename.

Introduction

Particles and Waves (Interferometer experiments with photons, particles and waves).
Clicking Single photon experiments under TOPICS can help navigation. You may instead use the Flash version with almost identical functionality (but which will run on fewer platforms). An o?ine version of the html5 simulation is provided for your convenience on Learn. Please indicate which version you use in your report, as the implementation of electromagnetic waves appears to be di?erent in the two versions.

 

You might find exploration of some of the other simulations in the Single photon experiments section and articles in refers to a photon in the upper path of the interferometer, not a horizontally polarised photon, and 0 1 ! refers to a photon in the lower path of the interferometer, not a vertically polarised photon,

 

Questions
1. Experiment
(a) Click on the Classical particles button and send at least 100 particles through.
[Note the Fast forward 100 counts button.] With what probability does the particle reach each detector? Does the coincidence counter record any counts?
(b) Click on the Electromagnetic wave button. The Phase shift in lower path slider will now appear. Fire at least 100 counts through for each of the 11 phase shift settings between 0 and ? and record the number of counts at each detector, and at the coincidence counter, as a function of phase shift.
(c) Click on Single photons. For each phase shift fire at least 100 photons through the interferometer. Record the number of counts at each detector, and at the coincidence counter, as a function of phase shift.

 

2. Theory
(a) The phase relationships in this Mach-Zehnder interferometer (with both beamsplitters, without a phase shifter) are such that the paths leading to detector 2 interfere constructively (light in the upper path, reflected from mirror 1 into detector 2, and light in the lower path, reflected from mirror 2 and beamsplitter 2 into detector 2, both having amplitude +1

 

2), while the paths leading to detector 1 interfere destructively (light in the upper path, reflected from mirror 1 and beamsplitter 2 into detector 1, and light in the lower path, reflected from mirror 2 into detector 1, having amplitude +1 2 and 1 2 respectively).

 

The glass plate now introduces a phase shift  into the lower path. Show that the intensities in the detectors become 1 ± ei 2 2 =(cos2 2 , detector 2 sin2

 

2 , detector 1 . (1)
(b) Recall the binomial distribution: if an event (such as a photon reaching detector 1) occurs with probability p in each of N trials, the mean number of events is N p and the standard deviation in the number of events is pN p(1  p). Hence find an expression for the standard deviation in the fraction of photons reaching each detector as a function of .

 

3. Interpretation
(a) Interpret, compare and contrast your results for classical particles, electromagnetic waves and single photons. This should include a discussion of the coincidence counter.
(b) Discuss whether your experimental data agree with the theoretical prediction in 2 (a), taking the expected error bars into account.
(c) Find a change you can make to the setup to reveal the path a photon has taken. What e?ect would the phase shift now have on the detection probabilities2?