Graphs: Some questions require production of a plot. You are welcome (indeed encouraged) to produce the plot using a package (e.g. EXCEL) where appropriate. Part of the assigned mark will be for the accuracy of the plot, and a part for presentation (readability, use of units, title, axis labels, suitable choice of axis ranges, etc.). A “publication quality” plot will get full marks. The plot can be printed on a separate sheet and referred to (or cut out and stuck on the page). Be careful - some online graphical packages are poor.
Descriptions: Some question require you to “explain” or “describe”. Please be succinct. A sentence is usually sufficient to make a point clear. I will mark “overall understanding” - thus a correct statement will gain marks but an incorrect statement will lose marks
1. This question refers to a wave function, ψ(x), for a particle confined to the x-axis in the region x ≥ 0. Your individual ψ(x) is provided in column 1 of the data sheet.
(a) Write down the expression for the position probability density function associated with your wave function ψ(x). Determine N, explaining the physical basis of your method.
(b) Produce a single plot containing both your wave function ψ(x) and the position probability density function found in (a).
(c) Find the most probable position for the particle.
(d) Find the expectation value of the position of the particle.
(e) Find <xˆ2 >.
(f) Find the expectation value of the x-component of momentum of the particle. Justify your answer.
(g) Find <pˆ
(h) Given the statistical relationship for an observable A (?Aˆ) 2 = <Aˆ2 > − ˆ 2 show that Heisenberg’s Uncertainty Principle is satisfied.
In this question, you are asked to design a two-slits experiment and to predict what results you will obtain. The unusual feature of your experiment is that, instead of light or electrons, molecules will be used. Column 4 of the data sheet contains your personal molecule.
The molecules are all derivatives of phthalocyanine which is a fluorescent dye. Once the molecules have passed through the slits, they adhere to a transparent screen. The molecules are excited by light from a laser and the emitted fluorescent light flash is focused onto a CCD camera. The CCD camera has an array of 3000×3000 pixels with each pixel having a resolution of 10 nm.
You have access to a source of your molecules as a gas at a temperature of 800 K. You also have a novel two-slit grating with the slits separated by 100 nm. The width of each slit is 10 nm and the thickness of the diffraction grating is also only 10 nm.
i. Calculate the average kinetic energy per molecule of your molecules at 800 K.
ii. Calculate the de Broglie wavelength λ of your molecules at this energy and present your result in nanometers.
iii. Source and present the equation that allows you to determine the separation of diffraction peaks for particles of de Broglie wavelength λ. Determine an appropriate distance for your detection system from the diffraction grating.
(b) Draw a clear, neat and well-labelled schematic diagram of your two-slit experiment. Include dimensions where these have been provided for you and include other quantities that you have calculated which are relevant to your experimental design.
(c) Provide a neat, hand-drawn, fully-labelled diagram of the final diffraction pattern you would expect to see from your two-slit experiment. This should be in the form of a line plot with a scale along the horizontal axis.
(d) You are asked to produce a movie to show the development of your diffraction pattern. Illustrate the development of the diffraction pattern by presenting four frames of the movie at different stages of the development of the diffraction pattern.