Momentum of a Free Particle
Consider a free particle whose wave function is \(\psi(x) = A\sin(p_0x/\hbar)\),
Is this wave function an eigenstate of momentum?
What are the possible results of a measurement of the momentum?
Calculate the expectation value \(\langle p\rangle\) and uncertainty \(\Delta p\) of momentum.
Dispersion Relation of a Free Particle
For a 1-D free particle, whose wave function is \(\psi(x) = Ae^{ikx}\), plot its dispersion relation, namely: the energy as a function of wave vector \(k\). Note \(k\) can be positive or negative. (The dispersion relation will come back later in the course.)
Position and Momentum Commutation
Calculate the commutator of the position and momentum operators. Do this two ways:
using the position representation of the operators
using the momentum representation of the operators
Derivatives of the Gaussian
The normalized Gaussian function is of the form
\[f(x)=\frac{1}{\sqrt{2\pi}\sigma} \,e^{-\frac{(x-x_0)^2}{2\sigma^2}}\]
Find the first two derivatives of the Gaussian function, by hand.
Make a table describing where the signs of the Gaussian itself and the signs of its first two derivatives are positive and negative.
Use your table to describe the shape of the Gaussian function.