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.
Fig: 1
Fig: 2
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
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.
[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.
\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}
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.
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.