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PHYC30018 ASSIGNMENT for 2023

Worth 15% of your final mark. Due Thursday 30 March before 5pm.

Please submit your assignment through LMS.

You are not permitted to use ChatGPT or any other AI tool to complete this assignment. You must show all lines of algebra. You are permitted to discuss the assignment questions with other stu- dents, but if you do so then write down the names of those students in your submitted work.

(c) It can be shown that Gaussian integrals where a is a complex number such that Re(a) ≥ 0 are also well-defined. They are called Fresnel integrals. (A famous example is 一(一) e ′ias2 ds =尸 with a a real number.)

Now, suppose that the ψ(x) in  (b) above is the wave function for a free particle at an initial time t = 0.

Explain how you would go about computing the time-evolved coordinate- space wave function ψ(x, t) through use of the momentum-space wave func- tion φ(p) you computed in  (b)(ii) – show that you get a Fresnel type of integral that when evaluated produces ψ(x, t).   You are not expected to actually perform the evaluation.

(6 + 4) + (10 + 5) + 5 = 30 marks

Question 2.

We are going to generalise the harmonic oscillator from one dimension to two dimensions x1  and x2 . The Hamiltonian is given by

Hˆ = 一 一 + m ω 1(2) x1(2) + m ω2(2) x2(2) + m ω1(2)2 x1 x2 .

When ω 12 0, we say that these two oscillators are coupled.

(i) For uncoupled oscillators,  use separated-variables functions of the form f (x1 )g(x2 ) to solve the energy eigenvalue problem. You may use the known result for the one-dimensional oscillator.

(ii) For the coupled case, the potential energy may be expressed in matrix notation as

V (x) = m╱x1    x2 、╱ ω 122(ω 12)     ω22(ω 122)  \ ╱x(x)2(1)  \ .

Taking inspiration from this, figure out a procedure for solving the general, coupled case.  (You are not expected to actually do all of the algebra, just enough of it to explain the method.)

15 + 15 = 30 marks