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Quantum Physics

(2) Consider group velocity dispersion in a medium described by the Sellmeier equation:

(a) Derive expressions for the group index N and the group velocity dispersion coefficient D .

(b) Plot the wavelength dependence of n, N and D, for fused silica in the region between0.25 and 3.5 µm, and for GaAs in the region between 1.5 and 10.5 µm. Compare and contrast their dispersive properties. Use the parameters provided in Table 2.1 (Module 2,slide 27 of the course notes; i.e., Saleh & Teich, 2nd ed., p.180).

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Quantum Physics

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?

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Quantum Physics

a) Show that the particle in a box wavefunctions w = Nsin(ntx/L) with 0 < x <L are orthogonal to each other.

b) Write a superposition wavefunction for the particle in a box. Why do we write superposition wavefunctions and not just eigenstate wavefunctions?

\text { Show that ground state wavefunction of the } \mathrm{H} \text { atom, } \psi=N \exp \left(-r / 2 a_{0}\right) \text {, is orthogonal to the } 2 \mathrm{~s} \text { orbital }

\psi=N\left(2-r / a_{0}\right) \exp \left(-r / 2 a_{0}\right) \text { and the } 2 \mathrm{p}_{\mathrm{x}} \text { orbital } \psi=N x \cdot \exp \left(-r / 2 a_{0}\right)

Explain why orthogonality is important in creating superposition wavefunctions.

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Quantum Physics

a) Briefly describe the relationships between the probability, probability density and probability 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.

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Quantum 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}

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Quantum Physics

7. In your own words, describe what "damping" does to a wave.

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Quantum Physics

6. Now, turn the damping up to "lots" and move the wrench up and down. In a couple of sentences describe what you see happening to the wave as it travels down.

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Quantum Physics

5. Switch to "Manual". Change the speed from “normal" to “slow motion". Move the wrench up and down and watch how the wave travels towards the clamp. In a couple of sentences describe what you see happening to the wave as it travels.

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Quantum Physics

4. Based off of Your Observations, create a definition for each of the terms below. (It might help if you draw a picture just for your own records)

Wavelength-

Frequency-

Amplitude-

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Quantum Physics

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

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Quantum Physics

2. Play with the Frequency in the bottom box. What do you see happen when you increase the frequency?

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