Advanced Quantum Mechanics: verschil tussen versies

Huidige versie van 26 dec 2022 09:24

About the course

This course is taught by Tjonnie Li since 2022. He is a nice prof that goes over the material in a good pace. He uses Sakurai as a reference book and this should be sufficient to do the exam. He also gives a mock exam in november to get an idea how the exam looks like. This mock exam does not do count towards the grade.

Before 2022 The course was taught by Christian Maes. He tends to be a bit vague, however still informing. The topics are material that were mostly covered in the bachelor, nevertheless you will notice that it is needed. Among the new topics are density matrices, decoherence scattering, quantum optics, a bit of quantum information theory and foundational issues. In November there is a test which does not count for the exam.

Academic year 2020-2021

Maes did go over some things he could ask. Here is a summary.

Summary Exam question

November test

Test 4 November 2020

Solutions of test 4 November 2020

20 January

Problem 1: True or False
- Every quantum algorithm can be simulated on a classical computer.
- A gate set ${U_{1}, \dots, U_{m}}$ , where each $U_{i}$ acts on a single qubit can never be a universal gate set.
- The Bell tests show that there cannot exist a hidden variable.
- The amplitude of Rabi oscillations is constant (in principle). (The point was in Rabi model it was true, but in Jaynes-Cummings not.)
- The dimension of the Hilbert space of five spin 1 particles is 243.
- The Aharonov-Bohm effect applies to neutrons and their interference pattern as well.
- ...
Problem 2: Show
- Show how you go from the first formula on page 42 of the lecture notes to the second one.
Problem 3: Calculate
- Consider a system composed out of three particles described by the state $α | 0 ⟩ | 0 ⟩ | 0 ⟩ + β | 0 ⟩ | 1 ⟩ | 0 ⟩ + γ | 1 ⟩ | 1 ⟩ | 1 ⟩$ in the Hilbert space $ℋ_{1} \otimes ℋ_{2} \otimes ℋ_{3}$ , where $ℋ_{i} = ℂ^{2}$ with basis ${| 0 ⟩, | 1 ⟩}$ and $| α |^{2} + | β |^{2} + | γ |^{2} = 1 .$
- Calculate $Tr (e^{σ_{x}} σ_{z} e^{- 2 σ_{z}})$ . The Pauli matrices were given.
- Variational principle to approximate ground state of a harmonic oscillator. The Hamiltonian is $H = \frac{P^{2}}{2 m} + \frac{m ω^{2}}{2} Q^{2}$ . The (still unnormalised) wavefunctions that you should use are of the form $ψ_{α} = e^{- α | x |}, α > 0$ . Compare with the exact solution.
Problem 4: Scatter:
- Differential cross section of bosons and fermions: difference with classical scattering, difference between bosons and fermions (look at 90 degrees), make a sketch of the scattering cross section and discuss briefly.
Problem 5: Perturb

Something similar as the following.

Problem 6: Fun
- How would you solve the measurement problem? Or perhaps you do not believe there is a measurement problem at all?
- What are the names of the authors of the EPR paper?

Academic year 2019-2020

Exam 2020 English translation:

1) Oral: A. discuss about symmetries of the Schrödinger equation: when is $[H, P] = 0$ ? What is rotation invariance? B. mixed states vs superposition of qubits: difference, give example of density matrix, give example of entangled state, calculate its density matrix, is the light in this room a mixture? C. scattering of identical bosons/fermions: difference with classical scattering, difference between bosons and fermions (look at 90 degrees), make a sketch of the scattering cross section.

2) The quantum circuit of the Toffoli gate. What does this gate do?

3) Subaddiditivity: prove that the photon number of coherent states is Poisson distributed.

4) A. Show how you go from the first formula on page 42 of the lecture notes to the second one. (Rewrite the Hamiltonian using an identity of Pauli matrices, and in terms of the magnetic field). B. Calculate $Tr (e^{- σ_{x}} σ_{z} e^{2 σ_{x}})$ .

Academic year 2016-2017

Exam 23 August 2017

Exam 25 August 2017

Academic year 2015-2016

Other

You can always look at the old course Advanced topics in QM.

@@ Regel 1: / Regel 1: @@
 [[Categorie: mf]]
 ==About the course==
-The course is taught by Maes. He tends to be a bit vague. The topics are weirdly mostly things you have already seen in the bachelor. Among the new topics are density matrices, scattering, a bit of quantum information theory and foundational issues. In November there is a test which does not count for the exam.
+This course is taught by Tjonnie Li since 2022. He is a nice prof that goes over the material in a good pace. He uses Sakurai as a reference book and this should be sufficient to do the exam. He also gives a mock exam in november to get an idea how the exam looks like. This mock exam does not do count towards the grade.
+Before 2022
+The course was taught by Christian Maes. He tends to be a bit vague, however still informing. The topics are material that were mostly covered in the bachelor, nevertheless you will notice that it is needed. Among the new topics are density matrices, decoherence scattering, quantum optics, a bit of quantum information theory and foundational issues. In November there is a test which does not count for the exam.
@@ Regel 13: / Regel 18: @@
 [[Media:TTT4nov2020solutions.pdf| Solutions of test 4 November 2020]]
+===20 January===
+#Problem 1: True or False
+#* Every quantum algorithm can be simulated on a classical computer.
+#* A gate set <math> \{U_1, \dots, U_m\} </math>, where each <math> U_i </math> acts on a single qubit can never be a universal gate set.
+#* The Bell tests show that there cannot exist a hidden variable.
+#* The amplitude of Rabi oscillations is constant (in principle). (The point was in Rabi model it was true, but in Jaynes-Cummings not.)
+#* The dimension of the Hilbert space of five spin 1 particles is 243.
+#* The Aharonov-Bohm effect applies to neutrons and their interference pattern as well.
+#* ...
+#Problem 2: Show
+#* Show how you go from the first formula on page 42 of the lecture notes to the second one.
+#Problem 3: Calculate
+#* Consider a system composed out of three particles described by the state <math> \alpha|0\rangle|0\rangle|0\rangle+\beta|0\rangle|1\rangle|0\rangle+\gamma|1\rangle|1\rangle|1\rangle </math> in the Hilbert space <math> \mathcal{H}_{1} \otimes \mathcal{H}_{2} \otimes \mathcal{H}_{3} </math>, where <math>\mathcal{H}_{i}=\mathbb{C}^{2} </math> with basis <math> \{|0\rangle,|1\rangle\} </math> and <math> |\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}=1. </math>
+#* Calculate <math> \text{Tr} \left( e^{\sigma_x} \sigma_z e^{-2\sigma_z}\right) </math>. The Pauli matrices were given.
+#* Variational principle to approximate ground state of a harmonic oscillator. The Hamiltonian is <math> H = \frac{P^2}{2m} + \frac{m \omega^2}{2}Q^2 </math>. The (still unnormalised) wavefunctions that you should use are of the form <math> \psi_\alpha = e^{ - \alpha|x|}, \alpha > 0 </math>. Compare with the exact solution.
+#Problem 4: Scatter:
+#* Differential cross section of bosons and fermions: difference with classical scattering, difference between bosons and fermions (look at 90 degrees), make a sketch of the scattering cross section and discuss briefly.
+#Problem 5: Perturb
+Something similar as the following.
+[[Bestand:Excercise.png]]
+#Problem 6: Fun
+#* How would you solve the measurement problem? Or perhaps you do not believe there is a measurement problem at all?
+#* What are the names of the authors of the EPR paper?
 ==Academic year 2019-2020==
 [[Media:131988018_193275205831807_5620500613001493258_n.jpg | Exam 2020]]
+English translation:
+) Oral: A. discuss about symmetries of the Schrödinger equation: when is <math> [H, P] = 0 </math>? What is rotation invariance? B. mixed states vs superposition of qubits: difference, give example of density matrix, give example of entangled state, calculate its density matrix, is the light in this room a mixture? C.  scattering of identical bosons/fermions: difference with classical scattering, difference between bosons and fermions (look at 90 degrees), make a sketch of the scattering cross section.
+) The quantum circuit of the Toffoli gate. What does this gate do?
+) Subaddiditivity: prove that the photon number of coherent states is Poisson distributed.
+) A. Show how you go from the first formula on page 42 of the lecture notes to the second one. (Rewrite the Hamiltonian using an identity of Pauli matrices, and in terms of the magnetic field). B. Calculate  <math> \text{Tr} \left( e^{-\sigma_x} \sigma_z e^{2\sigma_x}\right) </math>.
 ==Academic year 2016-2017==

Advanced Quantum Mechanics: verschil tussen versies

Huidige versie van 26 dec 2022 09:24

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About the course

Academic year 2020-2021

November test

20 January

Academic year 2019-2020

Academic year 2016-2017

Academic year 2015-2016

Other

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Advanced Quantum Mechanics: verschil tussen versies

Huidige versie van 26 dec 2022 09:24

About the course

Academic year 2020-2021

November test

20 January

Academic year 2019-2020

Academic year 2016-2017

Academic year 2015-2016

Other

Navigatiemenu

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