Colorado Mesa University

notes 2 (intro to QM and probability)

notes 3 (normalization, operators, and expectation values)

notes 4 (stationary states, infinite square well, and general solutions)

notes 5 (harmonic oscilator - algebraic method)

notes 6 (free particle)

notes 7 (delta function)

notes 8 (step potentials)

notes 9 (Formalism and linear algebra 1)

notes 10 (Formalism and linear algebra 2)

notes 11 (Formalism - inner products and operators)

notes 12 (Formalism - discrete, and continuous eigenvalues, statistics)

notes 13 (Formalism - Completed)

notes 14 (3D QM)

notes 15 (Hydrogen Atom Part 1)

notes 16 (Hydrogen Atom Part 2 Angular Momentum)

1.3--- Be careful with your U substitution, U = x-a, this changes the form for expectation values, check your math before you resort to looking up the integral.

1.4--- Psi is not the pdf, Psi^2 is, you'll end up with A^2, not A. Set up the integral for part e then use an integrator to solve it, it is tedious.

1.9--- Don't forget to properly sandwich your operators, look up integrals as needed, the expectation value of p^2 took me 15-20 lines WITH using the canned formulas for gaussian integrals. Tricks will make me assume you used the solutions, work it out until you can put it in a form to use tables or formulas.

1.10--- Do this with Excel, print out your worksheet and staple it to your homework.

1.17--- Part c is subtle, we only have psi(x,0), we'd need to calculate the expectation value of psi(x,t) to use it to get the expectation value for momentum.

2.4--- Use tables for things reduced to x/x^2*cos(x)

2.5--- For part c, reduce the Psi^2 to terms with sines and cosines then use tables. For part d use Ehrenfest's Theorem (see problem 1.7)

2.7--- This is an important one, it shows you how to expand any initial configurations in terms of basis functions, make sure you concentrate on this one, it is fundamental to QM. For part b, write out the integral explicitly then use an integrator (explain which integrator used and show how you get the series expansion on -1). Write down the formula for how you would do part d, I don't expect you to sum the series.

2.8--- Write down Psi(x,0) = A x element [0,a/2] 0 else. This should get you started.

2.13--- Don't forget the psi_ns are for the harmonic oscillator, don't forget conjugation in part b, use the raising and lowering operator formalism to get the result but don't read it off from 2.12, ask if clarification is needed. In part c use the time dependent form for psi. Also, since you are using the time dependent form, be smart about calculating the expectation of p. In part d use psi(x,0).

2.17--- Be smart about calculating derivatives, you've already done the n-1 ones.

2.21--- Use an integrator for part b.

2.22--- Reduce the integrals to forms you can use tables for, this takes more work than you might think. Be careful to "Bundle Up" groups of variables. Finding psi(x,t) requires two integral looks at equations 2.100 and 2.109. For d, express your integrals in terms of w and use tables. for p^2, skip it, tedious, the answer is hbar^2(a). For e, put in w explicitly and see if sigma_xsigma_p holds with the uncertainty principle.

2.26--- Use equation 2.102.

2.34-- Parts a and b were solved in class

2.38-- psi(x,0) = psi_1=sqrt(2/a)sin(pi x/a) for infinite square well. Psi_n are now basis functions where normalization constant is sqrt(1/a) and sin(n pi x/a) becomes sin(n pi x/ (2a)), expand psi(x,0) in terms of the new basis functions and get new c_ns to calculate probailities. Essesentially, your psi(x,0) now needs to be expanded in terms of basis functions where a goes to 2a.Be careful, this is two integrals. One is at n=2, the rest are not.

2.39-- e^i(x+2pi) = e^ix. Write the classical revival time in terms of energy.

4.22--- Let Y_l^l=e^(i l phi)F(theta).

4.24--- Write down the kinetic rotational energy of a system comprised of a massless rod joining two masses, express this in terms of angular momentum.

4.33--- The hamiltonian is time dependent, you must use the time dependent Schroedinger equation (4.162) to solve this. This will result in two time dependent equations to solve.

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