Extra Credit 44

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You have to master the following 10 problems and rewrite them to be unique from the original (2%). Then you have to solve the problem stepwise and thoroughly (2%) and in latex (1%).

McQuarrie and Simon Problems:

Q1.21
The Balmer series has a line with a wavelength of 656 nm. What is the energy of the electron?

Q1.18

When light of wavelength 318 nm shines on the surface of gold, electrons are ejected from the surface. The work function of fold is said to be 0.785 eV. What is the kinetic energy of electrons and the threshold frequency of gold.

Q2.18

Consider a mass suspended by a string. Assume the motion is such that it oscillates. Let's define x as the distance from equilibrium such that mds/dt=mldx/dt and the change of momentum is mld²x/dt²where m is the mass and l is the length of the spring. Show that the component of force in the direction of motion is -mgsin(x).

Now assume the distance from equilibrium is small so that sin(x) is about equal to x. What is the natural frequency of the harmonic oscillator?

Q3.18

Orthogonality is an extremely useful tool in quantum mechanics. Prove that the following set of functions is orthonormal

Y_n(x) = Asin(n*pi*x/L) where the bounds of x go from 0 to L.

Q4.17

Show that the following either commute or do not commute and explain why.

[L²,L_x]

[L²,L_y]

[L²,L_z]-

[L_y,L_x]

Q5.32

Convert the gradient vector from Cartesian to spherical coordinates and then find the square of the gradient vector in spherical coordinates.

Q7.13

Using the trial funciton Y_n(x) = k³- x³and the Hamiltonian of the harmonic oscillator, determine the ground state energy of the system. Is this a good approximation to the actual ground state energy of the harmonic oscillator?

Q8.20

Show how the time independent portion of the H atom equation is derived from the original time-independent equation.

Q9.14

Predict the stabilities of Ne, Ne⁺and Ne^2-.