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**Q1:**Write down the general expression for the expectation value of an operator Ô for aquantum mechanical particle described by a wave function V(x, t). Briefly describe theconnection between operators, wave functions, expectation values and experimental mea-surements.See Answer**Q2:**Question 4. A particle of mass m incoming from the left hand side is scattered from the double delta-function potential barrier V(x)=\frac{\alpha \hbar^{2}}{m}[\delta(x-a)+\delta(x+a)], \quad \alpha>0 . \text { where } \beta=1+2 i \alpha / k \text { and } \gamma=e^{i k a} \text { with } k=\sqrt{2 m E} / \hbar \text {. } Draw the scattering diagram and show that the transmission coefficient can be written as T=\left|\frac{4}{(1+\beta)^{2}-\gamma^{4}(1-\beta)^{2}}\right|^{2}See Answer**Q3:**2. (30 points) A quantum mechanical particle is in an eigenstate ) of Î² with eigenvalue 2ħ²: At a particular moment, the particle also is in an eigenstate of the x component of the angular momentum, Î, with eigenvalue 0. In other words, . Express this eigenstate of I, as a normalized superposition of the familiar eigenstates, |lm), of Î² and Îz, where Î¿ is the z component of the angular momentum.See Answer**Q4:**The spin operator in an arbitrary direction can be written as (0,0) = sin cos o + y sin sin + ₂ cos 0, I where the Pauli spin matrices are given by 0 (13), 1. Find the eigenvectors and eigenvalues for the operator ô(0, 6). ÔT x = Oy = 0 (² i). i 0 Oz = 0 0 -1See Answer**Q5:**2. Choose = 0 and find for an entangled state of the form 0 0 |I) >- (B),B),-),B)) 1/12 = 2 1 0 1 0 the probability of detecting particle 1 in spin-up with respect to an angle 0₁ and at the same time particle 2 in spin-up with respect to an angle 02.See Answer**Q6:**1. Starting from Maxwell's equations, derive a wave equation for the magnetic field and show that a harmonic plane wave prorogating along x direction is a solution. Prove it is a transverse wave. 2. Consider sunlight at the top of Earth's atmosphere. The sun-earth distance is 500 light-second. The sun's total power radiation is 3.9×1026 W. (1) What is the sunlight intensity? (2) What is the electric field amplitude? (3) How does its magnetic field amplitude compare to the Earth's magnetic field?See Answer**Q7:**3. A 800-nm pulsed laser emits one laser pulse every 13 ns. The duration of each pulse is 100 fs. You can treat each pulse as a squared pulse: That is, the power immediately jumps to its "on" value, stay constant for 100 fs, and then jumps back to zero until the next pulse. In reality, the pulses are more like a Gaussian function. The time-average power of the laser is 2 W. The beam size is 1 mm. Estimate: (1) Energy of each pulse. (2) Energy fluence of each pulse (that is, energy of each pulse per unit area). (3) Number of photons in each pulse. (4) Power, intensity, and electric field when the pulse is "on".See Answer**Q8:**4. A light wave is specified as following (in SI unites), Find: Ẽ =(−68 +3√5ŷ)(10¹) expil (√5x+2y)π×10² −9.42×10¹³t] . (1) The direction of electric field. (2) The scalar value of the amplitude of the electric field. (3) The direction of propagation. (4) The wave number and wavelength. (See Answer**Q9:**Problem 1: Exponential Switching Consider the following perturbation to a two-level system: Vfi (t)=x(1-e-at) e[t] Here x is the strength of the perturbation and a is the rate at which this strength is applied. In the plot below, the steepest curve is for the largest value of a. "Exponential Switching Figure 1 [a] Show that, in the limit of large time and infinitely slow exponential turn on, the transition probability as the following limit: x² w₂2 Use this expression is a useful asymptote that you will apply in parts [b] and [c] below. [b] Consider an exponential switching perturbation for which x = 0.1 w₁ and a = 0.1 w₁i. Plot the transition probability, using time steps of 1/ wf, and show the asymptote derived in [a]. Plt → arbitrarily large, a +0 = (1) (2) [c] Repeat the analysis of [b] but now with a much slower rate of turn-on: a = 0.001 wf. Plot the transition probability, using time steps of 1/ w, and show the asymptote derived in [a]. What is the difference between your results the two cases? Be sure o discuss the overall probability as well as the undulat in the transition probability with time. [d] Assume that the time-evolving state is a linear combination of the initial and final states. Substitute this into the Schrodinger equation, in the interaction picture, to obtain an ODE for M₁. Numerically solve this ODE to obtain p[t], and compare the result with your perturbation approximations of [b] and [c]. Comment on the accuracy of your perturbation approximation.See Answer**Q10:**Problem 3: Adiabatic Limit Re-consider the exponential switch perturbation, but now we will focus on extremely small values of the switch rate, a. This will lead us to an important theorem in Quantum Mechanics called the Adiabatic Theorem. The Adiabatic Theorem states that a system will stay in its evolving eigenstate provided the Hamiltonian is changed sufficiently slowly. Test this claim using the parameters of [1c]: x = 0.1 w₁ and a = 0.001 w₁₁. You already have the numerical and perturbative solutions to this transition. Now generate a third plot by directly calculating the occupation of the lowest eigenstate of H = Ho + V in the basis of Ho. This should allow you to calculate the occupation probability of the excited state of Ho as a function of time. Plot this probability along with your numerical and perturbative results, and comment on what you find. Aside: This is not the same notion of "adiabatic" as in thermal systems. Vaik ⓇSee Answer**Q11:**X 1. Light travels in a piece of glass with its electric field E₂ = E cos710¹5 (t- 0.65c angular frequency, wavelength, and index of refraction of glass, and its intensity. -). Find itsSee Answer**Q12:**2. Show that in a good conduction, (a) the skin depth of light is 2/27 and (b) the magnetic field lags the electric field by π/4.See Answer**Q13:**3. Consider a dielectric with a single resonance at @o. Calculate the width of the anomalous dispersion region (in which the index of refraction decreases with light frequency). Assume y<< Co. Show that the index of refraction reaches its extreme values at points where the absorption coefficient is at its half maximum.See Answer**Q14:**4. The full width at half maximum of an atomic absorption line at 589 nm is 100 MHz. A beam of light passes through a gas with an atomic density of 10¹7/m³. Calculate: (a) the peak absorption coefficient due to this absorption line. You can assume that the index of refraction is close to 1 in for this dilute gas; (b) the frequency at which the resonant contribution to the refractive index is at a maximum; (c) the peak value of this resonant contribution to the index of refraction.See Answer**Q15:**We know that a stationary state is of the form 4[r, t] = 4[r] e¹¹. (1) Here [] is an eigenstate and is the associated eigenenergy. (Remember that ħ= 1 in natural units.) In the idealized setting normally considered, this is a steady state of the system since it has a time-independent proba- bility density. In practice, though, all excited states have a finite lifetime, 1, and a more realistic representation of the probability density for any excited state is p[t] = e. (2) It is only through QFT that the decay of such "stationary states" are possible. With standard Schrödinger equa- tion quantum mechanics, a pragmatic expedient is to simply adopt a more physically reasonable excited state representation: 4[r, t] = 4[r] e\ อ้ 2 T t [a] Focus on the temporal component of this, T[t] :=et ²1, and calculate the following: (3) (i) Ť:= F[T], the temporal Fourier transform of T[t]. Call the Fourier frequency &, so that you have Ť[ɛ], a complex-valued energy spectrum for the wave function. (ii) The spectral density, D, is defined as D[ɛ] := Ť[ɛ] + Ť[ɛ]* = 2 Re[†[²]. Interestingly, the inverse Fourier transform of D[ɛ] is equal to T[t], so all we have really done is found a real-valued Fourier transform of the time-varying portion of the wave function. If you want, you can test this for yourself by calculating F-¹[D].See Answer**Q16:**[b] Plot the spectral density, D[], for a fixed value of & and several different values of T. (Put these all on the same plot, carefully labeling everything.) These plots should make clear that a finite lifetime, 1, implies a finite line width--i.e. a spectrum that peaks at & but is spread out in a Gaussian-like distribution around this energy as shown below./nKDmax A SE Dmax 2 Verify that the width of the distribution, is equal to 1/7 at the half-height of the peak. This is typically inter- preted as the range of energies that you might expect to measure in an experiment. It implies that τ δε = 1. (4) This is called the Lifetime Broadening Relation, and it should call to mind the time-energy uncertainty relation. Explain what the LB Relation says about the certainty with which you can know the excited state energy as a function of the lifetime of the excited state. Look at the two extremes, zero lifetime and infinite lifetime, to help elucidate the physics.See Answer**Q17:**#1 Calculate the radius of the first (n=1) Bohr orbit for the hydrogen atom. (10 points) Mass of an electron, m = 9.1e-³¹kgSee Answer**Q18:**#3 a) Calculate the energy in Joules of a green laser (wavelength of 530 nm) in a vacuum. (5 points) b) Convert the answer to energy in electron volts (eV) (5 points)See Answer**Q19:**#2 Calculate the energy of the first (n=1) Bohr orbit for the hydrogen atom in eV (10 points)See Answer**Q20:**#4 Calculate the wavelength of an electron in the ground state for a Hydrogen Atom (10 points) Velocity = 2.2X106. and mass = 9.1X10-³¹ kg Would you expect this electron with a diameter of 1.0X10-10 m to exhibit wave behavior? Why?See Answer

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