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:
A particle in an infinite square well potential has an initial state vector \[\left|{\Psi(0)}\right\rangle = A\big(\left|{\phi_1}\right\rangle -\left|{\phi_2}\right\rangle +i\left|{\phi_3}\right\rangle \big)\]
where \(|\phi_1\rangle,|\phi_2\rangle\), and \(|\phi_3\rangle \) are the first three energy eigenstates.
Determine \(A\).
At time \(t=0\), what are the possible outcomes of a measurement of energy, and with what probability would each possible outcome occur?
What is the average value of energy one would measure at \(t= 0\)? In other words, what is the expectation value of energy at \(t= 0\) ?
What is the quantum state of this particle at some later time \(t\)?
For a particle in an infinite square well from \(0\) to \(L\), calculate the probability of finding the particle in the range \(\dfrac{3 L}{4}<x<L\) for each of the first three energy eigenstates.
Consider an infinite square well potential between \(0\) and \(L\).
Write down an expression for the nth energy eigenstate.
Find the expectation value of position for the nth energy eigenstate.
Find the uncertainty of position for the nth energy eigenstate.
Find the expectation value of momentum for the nth energy eigenstate.