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Recently Asked Quantum Physics Questions

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Q1:3. A pendulum with length 1 is attached to the ceiling. It has an attached mass m. The robe is massless. We want to obtain the differential equation that describes the motion (no need to solve it). Make sure to write down all intermediate steps. (a) Use the Newtonian framework to obtain the differential equation describing the motion. (b) Use a single generalized coordinate, and use the Lagrangian framework to obtain the differential equation describing the motion. Indicate the zero of potential energy. (c) Use Cartesian x and y (pointing down) coordinates and the constraint of the pen- dulum rod to derive the equations of motion using Lagrange multipliers. (d) Use a single generalized coordinate and the Hamiltonian framework to describe the motion. Explicitly show that you can combine your first-order differential equations to obtain the same equations of motion.See Answer
Q2:13.5** A bead of mass m is threaded on a frictionless wire that is bent into a helix with cylindrical polar coordinates (p, o, z) satisfying z = co and p = R, with c and R constants. The z axis points vertically up and gravity vertically down. Using as your generalized coordinate, write down the kinetic and potential energies, and hence the Hamiltonian H as a function of and its conjugate momentum p. Write down Hamilton's equations and solve for and hence z. Explain your result in terms of Newtonian mechanics and discuss the special case that R = 0.See Answer
Q3:1. Solve the time-independent Shrodinger equation with appropriate boundary conditions for an infinite square well centered at the origin [V(x) = 0 for-a/2<x<a/2; V(x) == otherwise]. Confirm that your solution can be obtained by substitution x→x-a/2 in the results obtained in lecture notes. Calculate <x>, <x²>, <p>, <p²>, o, and op for the nth stationary state of the infinite well. Verify that the Heisenberg uncertainty principle is satisfied. Find the state that comes closest to the uncertainty limit. 2. Plot the Kronig-Penney solution over up to 4 periods for B=mavo/h²k-5 and Vo/a = 1. Find the energy at the top of the first allowed band and the energy difference of the band gap. Also show that the solution below Yn+1(x) = C[sin(Kx) + e¯ika sin(Ka)]; 0≤x≤a satisfies the Kronig-Penney model equation.See Answer
Q4:1) Normalize the following wave functions and evaluate the value of A. Sketch the probability density for each. (You can solve integrals by hand or by using a numerical tool. If you use a numerical tool, mention this in your answer.) a) ipx 4(x) = {Ae**, Ae , 0<x<L 0, otherwise b) (x) = Ae-x² c)(x) = Ae-2|x|See Answer
Q5:6. This problem will explore Blackbody radiation and its relationship with thermodynamics. As you may remember form class, Planck's law for blackbody radiation is: 8AV² hv 8 TV² (E)= c³ hv/kgI -1 u(v, T) = - The first term (²) represents the density of electromagnetic waves (harmonic oscillators) in the cavity and the second term, (E), represents the average energy of per oscillator. 1 1-e* a. Show how quantization of energy gives rise to Planck's Law. Hint: Start with quantized harmonic oscillators of the form Ennhv and calculate the average energy using the Boltzmann equation. You will need the following equations: Σex 00 Σ ne = d dx 71-0 nx 1 = -√√1-e²) d et (1-e² b. Show that at a given temperature there will be a most probable wavelength given by: Amax T = 2.898 x 10³ m/K (This is known as Wien's law and determines the color of a black body at a given temperature). c. Show that in the high and low temperature limits the Plank distribution becomes the Rayleigh- Jeans u(v, I) = 8²k₂I and Wien Lawsu(v, I) = 8 exp(r) respectively.See Answer
Q6:4. The Foerster energy transfer rate, kr, between donor and acceptor fluorophores was determined to be the following in 1946: 6 Ro -- ()* TD where to is the fluorescence decay time of the donor fluorophore, R is the actual distance from the donor absorption and acceptor emission electronic dipoles and R, is the critical distance (or Foerster Radius) at which the probability is that half of the excited state donor electrons will transfer their energy to the excited state of the acceptor. The critical distance (or Foerster Radius) can be calculated by 2108 [x²yn-47]1/0 Ro=0.2108 where is the orientational alignment factor between the donor absorption and acceptor emission electronic dipoles, D, is the fluorescence quantum yield of the donor, n is the refractive index of the media and J is the overlap integral or sum given by J = - FD(X)A(X)X²dX where Fo(A) is the wavelength dependent donor fluorescence, E(A) is the wavelength dependent acceptor absorption coefficients and A is the wavelength. For this exercise, please calculate the Foerster Radii for the following combinations of fluorophores for FRET pairs: Donor Cerulean (CFP) Acceptor mCitrine (Y pet) (YFP) To do this you will need to download the absorbance (or excitation) spectra and emission spectra of the fluorophores. You may find your own source, but it is suggested that you go to the Roger Tslen group website to download the fluorophore spectra. You will also need to obtain the fluorescence quantum yields of the donor fluorophores and the maximum absorption coefficients of the acceptor fluorophores. It will be fine to assume that the orientation factor will be random, which sets the = 2/3=0.6667, and that the refractive index will be that of water. You may use Mathematica, MATLAB, or even an MS Excel spreadsheet (heaven forbid!) to calculate these values. Please also note that the integration of Fo(A)dA is supposed to yield one. Thus, it might be necessary for you to determine the sum or integration of Fo(A)d over the spectral range and normalize the overlap integral J to this. It is important that you submit your work of your answers and notebook/workbook files/papers to show how you determined your answer.See Answer
Q7:3. Answer the following questions and be sure to show your work for each answer. • Identify the types and number of the symmetry elements and determine the point groups of the following molecules. • Use the character tables of the point groups to determine if the 1st and 2nd electronic transitions from the ground state is allowed or not for each of the molecules. a: H₂O b: NH3See Answer
Q8:2. Given the 4 level photophysical-system mentioned during the lecture, with a singlet ground state, singlet excited state, triplet excited state, dark state, and respective rate constants (defined in lecture and shown on the diagram below), determine: a. the fluorescence emission rate, F, dependence on the other parameters in a single equation and graph the fluorescence emission rate dependence on the excitation intensity, / (i.e., fluorescence emission rate, F, on the y-axis and excitation intensity, I, on the x-axis) b. the maximal fluorescence emission rate when the excitation intensity approaches infinity (i.e., when /→ ∞, F=?) c. the saturation intensity, Is no Kerans ko $0 n k = ol ng 2 Kisc nSee Answer
Q9:Exercise 2 (16 points): Quantum Phase Estimation The phase estimation algorithm is a building block of many other quantum algorithms. Consider a circuit that implements the single-qubit gate U, and a single-qubit state) which is an eigenstate of U. 1. (1 point) Prove you can write U)²) where is a real number between 0 and 1. 1 The phase estimation algorithm is a method to find when a circuit performing U and a qubit in state ) are given. Specifically, we will be performing the following circuit: Superposition Controlled U Operations Measurement 10) H 10) H 间 H QFT 10) Each of the top n qubits will end up containing a digit of the binary floating point decomposition of the real number 9. We can write 20-a+2º5, where a is the nearest integer to 2º0 and 6 is the "rounding error" in the n-bit representation of 6, i.e. [26] ≤ 4. We will assume 5-0 in this homework. Notice the need for controlled U gates, i.e. conditional on a control qubit, perform the gate U on a target qubit 1 times. We also need the Inverse Quantum Fourier Transform for which we know QFT-¹ – QFT↑ (because the QFT is unitary). 2. (1 points) Write down in terms of projectors and U matrices the two-qubit gate Ctrl(U) (the gate that controlled on the first qubit applies the gate U on the second qubit). 3. (3 points) Write down the gates Ctrl(U²) and (Ctrl(U))² and prove they are the same gate. The first gate stands for "conditional on the control qubit perform U² on the target qubit". The second gate stands for repeating twice the operation "conditional on the control qubit perform U on the target qubit/nEach of the top n qubits will end up containing a digit of the binary floating point decomposition of the real number 9. We can write 20-a+2", where a is the nearest integer to 20 and 6 is the "rounding error" in the n-bit representation of 6, i.e. [26] ≤ 4. We will assume 5-0 in this homework. Notice the need for controlled U gates, ie. conditional on a control qubit, perform the gate U on a target qubit 1 times. We also need the Inverse Quantum Fourier Transform for which we know QFT-¹ – QFT↑ (because the QFT is unitary). 2. (1 points) Write down in terms of projectors and U matrices the two-qubit gate Ctrl(U) (the gate that controlled on the first qubit applies the gate U on the second qubit). 3. (3 points) Write down the gates Ctrl(U²) and (Ctrl(U))² and prove they are the same gate. The first gate stands for "conditional on the control qubit perform U² on the target qubit". The second gate stands for repeating twice the operation "conditional on the control qubit perform U on the target qubit 4. (1 point) What is the state of the first n qubits immediately after the H gates? 5. (4 points) What is the state of all qubits just before the inverse QFT? Is the system that started in state) factorizable out or is it entangled with the rest of the qubits? 6. (4 points) What is the state of the top n qubits after the inverse QFT? Answer this question in the basis of bitstrings (ie. the Z basis). 7. (2 points) When we measure the top n qubits in the Z basis, what is the chance of observing the result to be the integer a expressed in binary? 2 Notice that finding the phase to n bits of precision required (2") applications of controlled-U. In other wordsto achieve precision we need (1/c) controlled gates. As a very rough comparison to classical estimation, consider the following nearly equivalent problem: you have a coin that with probability p gives heads, otherwise gives tails; you will need (2) samples to get varience on the order of, Le. error on the order of estimation. Le bits of precision with a classicalSee Answer
Q10:8.9 ** Consider two particles of equal masses, m₁ = m2, attached to each other by a light straight spring (force constant k, natural length L) and free to slide over a frictionless horizontal table. (a) Write down the Lagrangian in terms of the coordinates r₁ and r2, and rewrite it in terms of the CM and relative positions, R and r, using polar coordinates (r, ) for r. (b) Write down and solve the Lagrange equations for the CM coordinates X, Y. (c) Write down the Lagrange equations for r and p. Solve these for the two special cases that r remains constant and that remains constant. Describe the corresponding motions. In particular, show that the frequency of oscillations in the second case is w=√2k/m1.See Answer
Q11:Pom the separation ro. 8.14*** Consider a particle of reduced mass μ orbiting in a central force with U = kr" where kn > 0. (a) Explain what the condition kn > 0 tells us about the force. Sketch the effective potential energy Ueff for the cases that n = 2, -1, and -3. (b) Find the radius at which the particle (with given angular momentum () can orbit at a fixed radius. For what values of n is this circular orbit stable? Do your sketches confirm this conclusion? (c) For the stable case, show that the period of small oscillations about the circular orbit is Tosc = Torb/√√n +2. Argue that if √√n+2 is a rational number, these orbits are closed. Sketch them for the cases that n = 2, −1, and 7.See Answer
Q12:8.18** An earth satellite is observed at perigee to be 250 km above the earth's surface and traveling at about 8500 m/s. Find the eccentricity of its orbit and its height above the earth at apogee. [Hint: The earth's radius is Rê ≈ 6.4 × 106 m. You will also need to know GM, but you can find this if you remember that GM/R2 = g.]See Answer
Q13:5. Calculate the line integral of the function v = x² + 2yz ŷ + y² 2 from the origin to the point (1, 1, 1) by three different routes: (a) (0, 0, 0)→ (1, 0, 0) → (1, 1, 0) → (1, 1, 1) (b) (0, 0, 0)→ (0, 0, 1) → (0, 1, 1) → (1, 1, 1) (c) The direct straight line. (d) What is the line integral around the closed loop that goes out along path (a) and back along path (b)?See Answer
Q14:Task 1: Quantum Physics In this unit, you were introduced to and learned how to use the following equations: • Energy unit conversions between electron volts (eV) and joules, (J); Einstein's photon energy equation; . Compton's momentum equation; . de Broglie's wavelength equation Using your knowledge of these equations, quantitatively compare a 3.1 eV photon and a 3.1 eV electron by completing the following data table: Use the following Task 2 Data Table document to complete this task. Note: For any required calculation, be sure to include both your calculation and your answer. You may type out your answers, save your document as a PDF file then submit your completed work. Property Energy (J) Rest mass (kg) Speed (m/s) Wavelength (m) Momentum (kg-m/s) Photon Electron/nTask 2: Special Relativity 1. Suppose that you have been chosen for a space mission to a distant planet. Due to the length of time you'll be away from Earth you must carry out physical activity every day. On earth your, strength and conditioning trainer has determined you must do 90 minutes of exercise every day. If the vehicle is travelling at 0.80 o how much time, according to a timer on the space vehicle should you be active to meet your physical activity requirement? contart.ca-car-t.com-avegh.com 2. Suppose a space vehide with a rest mass of 150 000 kg travels past the International Space Station at a constant speed of 2.6 x 10 mis with respect to the 1.5.5. When an observer on the 1.5.5. messures the moving vehicle, her measurement of the space vehidle length is 250m a. Determine the relativistic mass of the space vehicle. b. Determine the length of the space vehicle as measured by an astronaut on the space Note: For any required calculation, be sure to include a complete solution process with a summary statement and a final answer with the correct significant figures. C ACTION CONSOLIDAN/nTask 3: Quantum Theory k=6.63 x 10Js; e-3.00 x 10° Vr: m,-9.11 x 10 kg 1. A potassium surface lo lluminated by a monochromatic laser light with a wavelength of 400 nm. Determine the maximum speed of a photoelectron emited from this surface the potassium surface has a work function of 2.30 ev. m,-9.11 x 10 kg 2. A photon ha a frequency of 7.50 x 10¹4 H a. Determine the energy and the momentum of this photon b. If all the energy of this photon were to be converted to mase, determine the equivalent mass for the particle. c. A microscopic specimen has a wavelength of 8.2 x 10m and a speed of 1.1 x 10 m/s. Determine the mass of this microscopic specimen. 3. In a TV tube an electric potential difference accelerates electrons froma rest poetion towards a screen. Just before reaching the screen, the electrons have a wavelength of 1.2 x 10m. Determine the kinetie energy of the electrone just before they reach the screen. Your final answer should have units of ev Note: For any required calculation, be sure to include a complete solution process with a summary statement and a final answer with the correct significant figures ning ACTION/nTask 4: Developments in Modern Physics 1. Two atomic clocks are synchronized. One is placed on a satelle, which is around the earth at high speed for a whole year. The other is placed in a lab and remains atrast, with respect to the earth. You may assume that both clocks can measure time accurately to many significant digits. a. Will the two clocks still be synchronized after one year? Explain your reasoning b. Imagine that the speed of light was much slower than its actual value. How would the results of this experiment change if the speed of light was only twice the average speed of the satelle? Explain your reasoning, using a calculation. 2. Use Max Planck's quantum theory to explain the following behaviour of photoelectrons. a Low-intensity light does not release any photoelectrons. What will happen if the light is made brighter? Explain your reasoning b. Low-intensity light releases photoelectrons. What will happen if the light is made brighter? Explain your reasoning. Low-intensity light does not release any photoelectrons. What will happen if the frequency of the light is gradually increased? Explain your reasoning The teacher will assess your work using the following rubric. Before submitting your assessment, review the rubric to ensure that you are meeting the success criteria to the best of your ability Submission instructions When you are ready, submit your assessment for feedback and a grade by selecting the "Assignment" ink and flowing the submission directions. Rubric Tak 1See Answer
Q15:7. (This is an optional question for bonus credit) Mixed unitary channels [6 points]. Prove that the channel in question 6 cannot be expressed as a mixed-unitary channel with only three Kraus operators. Note: If you submit a solution to this question then there is a size-limit of one page.See Answer
Q16:6. Mixed unitary channels [8 points]. A channel is mixed-unitary if it can be expressed in terms of Kraus operators Ao, , Am-1 that are of the form Ak = √Pk Uk, where Uo,...,Um-1 are unitary and (Po... Pm-1) is a probability vector. Consider the one-qubit channel that maps every 2×2 density matrix to the density matrix [1/2]. Show that this is a mixed-unitary channel by exhibiting four Kraus operators, A0, A1, A2, A3, for it that are in the form of a mixed-unitary channel.See Answer
Q17:5. Some mixed-state state distinguishing problems [8 points, 4 each]. Consider the following state distinguishing problems, where you are given the state on the left or the right (each, with probability) and your goal is to correctly guess which of the two states it is. In each case, give the maximum possible success probability possible and describe the measurement procedure that attains it: (a) (b) state (0) state (0) VS. VS. state state 1) with prob. with prob. 10) with prob. +) with prob.See Answer
Q18:4. A particular class of quantum channels [12 points; 4 each]. Define the qubit-to-qubit channels XA and XB as where Ao Во = noMRIENTERED ON WIR XA(P) = APA + A₁A₁ XB(P) = BopB + B₁B₁, 0 [6] [A] and 0 [₁] State your answer and justify it. and Co = A₁ = for parameters p, q = [0, 1]. (a) Confirm that A and A₁ are valid Kraus operators by calculating AA0 + A₁ A₁. B₁ = [8 VT-P] (b) Describe in words what the channel XA does in the case where p = 0. Describe in words what the channel XA does in the case where p = 1. (c) Consider the channel obtained by first applying XA and then applying XB. Denote this composed channel as XBO XA, where (XBO XA) (P) = XB(XA(P)). From the definition, the composed channel XBO XA has Kraus operators AoBo: AB₁, A₁B0, and A₁B₁. Write out the entries of these four matrices. Is XBOXA the same channel as Xc, with Kraus operators and C₁ (5) pq - [8 VT-P4] ? G= (6) (7) (9)See Answer
Q19:3. Detail about measurements associated with Kraus operators [8 points]. Prove that if matrices A₁, A2,..., Am are Kraus operators (i.e., they satisfy the equation Σk=1 A₁ Ak = I) and p is any valid density matrix then, for all k € {1,2,...,m}, AkpA Tr(AkpA) (3) is also a valid density matrix. (A square matrix M is a valid density matrix if, for all states ), it holds that (M) ≥ 0 and Tr(M)= 1.)See Answer
Q20:2. Action of unitary operations on the Bloch sphere [12 points; 4 each]. For every 2 x 2 unitary matrix U, the effect of applying U on a qubit can be viewed as a rotation of the states in the Bloch sphere. For example, it can be shown that (2) corresponds to rotation by angle 20 along the axis specified by state 10). In each case below, give the angle of rotation and also specify the axis of rotation: [cos (a) sin (b) (c) cos i sin cos sin sin 0] cos i sin t cos sin 0 cos 0 [o] CONTINUED ON NEXT PAGE 1See Answer

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