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PHOT 301: Quantum Photonics
Midterm exam questions (retake)
Michaรซl Barbier, Fall semester (2024-2025)
General information on the exam
Grading : This midterm exam will count for 10% of your total grade. Together with the projects which count for 40%, and the final exam that will account for 50%, your total grade for the course will be determined.
Exam type : The midterm exam consists of 7 open questions/problems. The exam is a written exam and all questions can be answered using only pen and paper. Calculators, mobile phones, laptops are not needed, and are not allowed to be used during the exam.
The duration of the midterm exam is 2 hours.
Exam questions
Please fill in all questions listed below. Each of the questions is valued equally in the score calculation of the exam.
Please tell if any question is unclear or ambiguous.
Question 1: Wave functions and expectation values
Consider the following wave function defined on ๐ฅ โ [0, 1]:
๐(๐ฅ) = ๐ด๐ฅ(1 โ ๐ฅ)^2
with ๐ด a normalization constant.
(1/3) First calculate the normalization constant ๐ด of the wave function. (2/3) Then calculate the expectation value for the position operator โจฬ๐ฅโฉ โ [0, 1]. (3/3) Afterwards calculate the expectation value for the momentum: โจฬ๐โฉ = โจโ๐โ (^) ๐๐ฅ๐ โฉ.
Hints : For a wave function ๐(๐ฅ) with ๐ฅ โ [๐, ๐], the expectation value for an operator ๐ฬ is given by:
๐
๐
๐(๐ฅ)โ^ ๐ ๐(๐ฅ)๐๐ฅ.ฬ
You also can make use of the following definite integral (here parameters ๐ and ๐ are integers):
1
0
๐ฅ๐^ (1 โ ๐ฅ)๐^ ๐๐ฅ =
where the factorial of a positive integer ๐! = ๐ โ
(๐ โ 1) โ
(๐ โ 2) โ
โฏ โ
2 โ
1.
Question 2: Ladder operators in a harmonic oscillator
(1/1) Prove the following equality:
[ฬ๐โ ๐+ฬ ๐ฬ+ + 2ฬ๐โ โฬ ๐+] ๐ 0 (๐ฅ) = ๐ 1 (๐ฅ)
with ๐๐(๐ฅ) the eigenstates of the harmonic oscillator with Hamiltonian:
๐ป = โ๐ (ฬ๐ฬ + ๐โฬ +^1
Hint : the commutator [ฬ๐โ,ฬ ๐+] = ๐ and the ladder operators acting on an eigenstate ๐๐(๐ฅ):
Question 3: Oscillations of eigenstates
For an infinite square well with width ๐ฟ the solutions can be written in the form:
sin (
โ^2 ๐^2 ๐^2
2๐๐ฟ^2
, with ๐ = 1, 2, 3, โฆ
Consider the wave function ฮจ(๐ฅ, ๐ก), which has the following form:
ฮจ(๐ฅ, ๐ก) = ๐ด (๐ 1 (๐ฅ)๐โ๐๐ธ^1 ๐ก/โ^ + ๐๐ 2 (๐ฅ)๐โ๐๐ธ^2 ๐ก/โ).
(1/2) Find the normalization factor ๐ด for the wave function. (2/2) Calculate the expectation value for the position โจฬ๐ฅโฉ as a function of time and show that it oscillates (in time) around the center of the well ๐ฟ/2.
Question 5: Projection operators
Consider a system with the following orthonormal basis of eigenstates:
Consider further the projection operator ๐๐ผฬ = |๐ผโฉโจ๐ผ| with:
(1/1) Perform the following projection:
Question 6: Commutators
(1/1) Calculate the commutator:
[ฬ๐ฅ^2 ,ฬ ๐^2 ] =ฬ ๐ฅ^2 ๐ฬ^2 โฬ ๐^2 ๐ฅ^2 ฬ,
with the position operator ๐ฅ = ๐ฅฬ and the momentum operator ๐ = โ๐โฬ (^) ๐๐ฅ๐.
Question 7: The spectrum of eigenvalues in matrix formalism
(1/2) Extract the eigenvalues for the following observable operator ๐ฬ in a system de- scribed in a two-dimensional vector space:
(2/2) Afterwards, find the corresponding eigenstates and normalize them.
Hint : Eigenvalues ๐๐ are given by the characteristic equation, i.e., you need to find ๐๐ such that the determinant of the matrix: det(๐๐๐ โ ๐) = 0.
The determinant of a 2 ร 2 matrix can be calculated as follows:
