Students do calculations for time evolution for spin-1.
Spin-1 Time Evolution
Use a completeness relation to rewrite each of the states below in the \(z\)-direction.
- \(|\psi_A\rangle = |-1\rangle_x\)
- \(|\psi_B\rangle = \frac{1}{\sqrt{2}}|+1\rangle_z - \frac{1}{\sqrt{2}}|-1\rangle_z\)
- \(|\psi_C\rangle = |-1\rangle_z\)
Suppose the Hamiltonian is given by \(\hat{H} = \omega_o \hat{S}_x\).
- Identify the energy eigenstates and their corresponding energy eigenvalues.
- Use them to write each state at a later instant in time.
- Represent each state using Arms as complex numbers.
For \(|\psi_C\rangle\):
List the possible values of a measurement of \(S_z\) and calculate the corresponding probabilities.
List the possible values of a measurement of \(S_x\) and calculate the corresponding probabilities.
- Graph any probabilities that depend on time.
accessibility_new Kinesthetic
10 min.
group Small Group Activity
30 min.
computer Mathematica Activity
30 min.
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Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.face Lecture
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assignment Homework
Consider the following wave functions (over all space - not the infinite square well!):
\(\psi_a(x) = A e^{-x^2/3}\)
\(\psi_b(x) = B \frac{1}{x^2+2} \)
\(\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)\) (“sech” is the hyperbolic secant function.)
In each case:
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Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.assignment Homework
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keyboard Computational Activity
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Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.group Small Group Activity
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central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.