量子物理代考-PHY356代写-quantum physics代写
量子物理代考

量子物理代考-PHY356代写-quantum physics代写

PHY356H1F

Duration: 3 hours

Aids Allowed: One handwritten 8.5″x11″ aid sheet (both sides)

量子物理代考 The exam is out of 100 marks. There are four questions. All questions are of equal value. Total Pages=6 (including cover sheet).

Exam Reminders:

  • Fill out your name and student number on the top of this page.
  • Do not begin writing the actual exam until the announcements have ended and the Exam Facilitator has started the exam.
  • As a student, you help create a fair and inclusive writing environment. If you possess an unauthorized aid during an exam, you may be charged with an academic offence.  量子物理代考
  • Tum off and place all cell phones, smart watches, electronic devices, and unauthorized study materials in your bag under your desk. If it is left in your pocket, it may be an academic offence.
  • When you are done your exam, raise your hand for someone to come and collect your exam. Do not collect your bag and jacket before your exam is handed in.
  • If you are feeling ill and unable to finish your exam, please bring it to the attention of an Exam Facilitator so it can be recorded before leaving the exam hall.
  • In the event of a fire alarm, do not check your cell phone when escorted outside.

Special Instructions:

Please read each question thoroughly before answering. Only answer questions you are asked!

Unsupported answers will only receive partial credit – show your work.

Exam Format and Grading Scheme:  量子物理代考

The exam has four multipart questions. All four questions are of equal value.

The exam is out of 100 marks. There are four questions. All questions are of equal value. Total Pages=6 (including cover sheet). 量子物理代考

Please read each questiiJn thoroughly before answering. Only answer questions you are asked!

Unsupported answers will only receive partial credit show your work.

Some possibly useful formulas are given at the end.

1.[25 marks]

(a) The Hamiltonian for a certain two-level system is

where {11), 12)} form an orthonormal basis. Find (i) the matrix representing Hin this basis, and (ii) the normalized stationary states of the system and their energies.

(b) A particle in one dimension is prepared in a state with the wave function

where ‘I/Jo is a real constant, and a is a positive constant.

i.Calculate ψo.

ii.Calculate

iii. Calculate the momentum space wavefunction (ψ,(px).

iv.Express 13.Px as an integral in momentum space, and show that it is infinite. Where does this infinity arise if instead 13.Px is written as an integral in position space?

量子物理代考
量子物理代考

2. [25 marks]  量子物理代考

(a) The ground state wavefunction for a particle in a simple harmonic oscillator potential

is

Use the raising operator to find the normalized wavefunction of the first excited state.

(b) At t = 0 a simple harmonic oscillator of mass m and frequency w is in the state

where the n’th excited state is denoted by In), co is a real constant, and .

i.Find co and lc1i if the expectation value of the energy in |ψ〉is

ii.Find Iii if, in addition,

量子物理代考
量子物理代考

iii. Find 〈X〉(t).

3.[25 marks]

(a) Show that the expectation value of Lx in any eigenstate of Lz vanishes.  量子物理代考

(b) The Hamiltonian for a three dimensional isotropic harmonic oscillator is

Explain in a few sentences why eigenstates of H can always be chosen to be of the form

and explain the physical significance of and m. (Your answer should include the reason that f(r) depends only onand E, not m, and that depends on m and,

not E. You don’t need to include explicit calculations in your explanation.)

( c) Suppose the wavefunction of the particle in part (b) has the form

where r =  and g(r) is some function of r. (i) What are the possible values of 2 that can be obtained from a measurement of the system and the corresponding probabilities? (ii) What are the possible values of Lz that can be obtained from a measurement of the system and the corresponding probabilities?

( d) Explain briefly why the recursion relation

implies that the energy levels of the electron in the hydrogen atom are discrete, not continuous. Use this relation to find the wave function for the lowest energy state of

hydrogen with £ = 2 and m = 1 as a function of r, () and ¢. You do NOT need to normalize it.

4.[25 marks]  量子物理代考

The Hamiltonian for a three-dimensional Hilbert space is represented in the basis

{|1〉, |12〉, |3〉} by the matrix

量子物理代考
量子物理代考

An observable A is represented in the same basis by the matrix  量子物理代考

At t = O, the system is prepared in the state

(a) Show that the stationary states of the system are

and find the energy of each state.

(b) Find |ψ(t)〉 in the {|1〉, |2〉, |3〉} basis.

( c) At time t the observable Λ is measured. What possible values can be obtained, and

what is the probability for each?

(d) At time to, A is measured to be a. Find |ψ(t)〉 in the {|1〉, |2〉, |3〉} basis for t > to.

 

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量子物理代考
量子物理代考

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