Three-D oscillator Consider a particle in a 3 -D oscillator potential Determine its ground state wavefunction and energy in iterms of the three oscillatorquantum

3) Consider a free particle wave packet for which the wave number distribution A(K) = { C for - Ak ≤ k ≤ Ak 0 elsewhere with constant C > 0. a) Determine the constant C so that A² (k) is normalized. b) Sketch A(k). c) Determine the associated spatial probability density (x, 0)|² at time t = 0. Determine the value of y(x,0)|² at x = 0. Use these results to sketch |(x,0)|². d) Approximating the width of A²(k) as Ak and the width Ax as the distance from the origin to the first minimum of y(x, 0)|², determine the product Ax Ap at time t = 0 and comment on your result. e) Write down an explicit integral expression for (x, t). You do not need to solve this integral!

5) If  is a Hermitian operator, then *  dx = f(¢) dx for wave functions and (optional proof available in Moodle in section 2). of a Hermitian operator  are orthogonal Using this relation, show that the eigenfunctions provided that they have different eigenvalues. Explain the significance of this result.

4) a) Determine the adjoint operator to â = c + id, where c and d are real numbers and i = √-1. Is â Hermitian? b) By evaluating (E) - (E) with respect to a general normalized wave function, determine whether or not the operator = this Hermitian. Interpret your result.

1. In the simulation, switch to "Oscillate". Play with the Amplitude in the bottom box. What do you see happen to the wave on the string when you increase the amplitude?

3. Click on the Rulers button on the bottom box. Then set the amplitude and frequency to the numbers below. After the numbers are set, play the wave and pause it after 5 seconds. Measure the distance between the top of one wave to the next wave. Record that as the Wavelength.Amplitude

b) Briefly describe how the general shape of the wavefunction can be described without actually solving the Schrodinger equation. c) Briefly explain the fundamental relationship between operators and physical observables in quantum mechanics. What is a physical observable and what kinds of operators are associated with physical observables? d) Describe the composition of a localized particle written as a superposition of momentum states(eigenfunctions of momentum) and how this composition changes as the particle becomes more localized. a) Briefly describe the relationships between the probability, probability density and probability amplitude.

\text { Consider the } \mathrm{X} \text {-force equilibriu } \mathrm{m}, \sum \mathrm{F}_{\mathrm{x}}=0 \text {; } Cancelling out equal terms in opposite faces \frac{\partial \sigma_{x}}{\partial x} d x d y d z+\frac{\partial \tau_{y x}}{\partial y} d x d y d z+\frac{\partial \tau_{z x}}{\partial z} d x d y d z+X d x d y d z=0 So we get the x- equilibrium equation \frac{\partial \sigma_{x}}{\partial x}+\frac{\partial \tau_{y x}}{\partial y}+\frac{\partial \tau_{z x}}{\partial z}+X=0 \frac{\partial \tau_{y x}}{\partial x}+\frac{\partial \sigma_{y}}{\partial y}+\frac{\partial \tau_{y z}}{\partial z}+Y=0 \frac{\partial \tau_{z x}}{\partial x}+\frac{\partial \tau_{z y}}{\partial y}+\frac{\partial \sigma_{z}}{\partial z}+Z=0 in short, the equilibrium equations in tensor notation Ou, +X, = 0 (i, j = x, y, z) Take moment quilibrium about an axis through the center and parallel to z - axis \tau_{x y} d y d z \frac{d x}{2}+\left(\tau_{x y}+\frac{\partial \tau_{x y}}{\partial x} d x\right) d y d z \frac{d x}{2}-\tau_{y x} d x d z \frac{d y}{2}-\left(\tau_{y x}+\frac{\partial \tau_{y x}}{\partial y} d y\right) d x d z \frac{d y}{2}=0 \tau_{x y} d x d y d z-\tau_{y x} d x d y d z=0 \therefore \tau_{x y}=\tau_{y x} \tau_{y z}=\tau_{z y} \tau_{z x}=\tau_{x z}

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.

Provide a physical explanation for the shape of the wavefunction for a particle in box with changes in particle mass and box width. Explain for the energy of the particle in box with changes in particle mass and box width. What causes the particle to have discrete (quantized energies)? Hint: The free particle Schrodinger equation is the same as the particle in a box wavefunction, but its energies are not quantized. You have non-interacting (ideal gas) molecules of mass m in a vessel of width L. You compress the vessel adiabatically (in QM, this means, you compress slowly so that the particles stay in their same quantum states as you compress). Explain how much work was done to compress the gas. Did the temperature change?

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.