Consider quantum particles in different 1-D potentials U(x).
For each of the following wavefunctions:
(i) Confirm that the wavefunction is normalizable by showing that the wavefunction goes to zero at \(\pm\infty\).
(ii) Find the value of the overall constant that normalizes the state over all space.
(iii) Calculate the probability of locating the particle in the region \(x = 0\) m to \(x = 1\) m.
(iv) (For the first two states \(\left|{\psi_1}\right\rangle \) and \(\left|{\psi_2}\right\rangle \) only) Calculate the probability that the particle will be in a state
\[\left|{\psi_{out}}\right\rangle \doteq \begin{cases} 0 & x < 0 \\ \sqrt{2}\sin \pi x & 0\leq x\leq 1 \\ 0 & x > 1 \end{cases}\]
after an unspecified measurement process.
(a) \(\left|{\psi_1}\right\rangle \doteq \begin{cases} 0 & x < -1 \\ A & -1\leq x\leq 1 \\ 0 & x > 1 \end{cases}\)
(b) \(\left|{\psi_2}\right\rangle \doteq \begin{cases} 0 & x< 0 \\ Bx(x-3) & 0\leq x\leq 3 \\ 0 & x > 3 \end{cases}\)
(c) \(\left|{\psi_3}\right\rangle \doteq \begin{cases} Ce^{k(x-1)} & x\leq 1 \\ Ce^{-k(x-1)} & x > 1 \end{cases}\)
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.
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 \(t\)?
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 \(|\phi_n\rangle\) are the energy eigenstates. You have previously found \(\left|{\Psi(t)}\right\rangle \) for this state.
Use a computer to graph the wave function \(\Psi(x,t)\) and probability density \(\rho(x,t)\). Choose a few interesting values of \(t\) to include in your submission.
Use a computer to calculate the probability of measuring the particle to be near the middle of the well (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.
Choose another location in the well, different from the location above. Use a computer to calculate the probability of measuring the particle to be near your chosen location (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.
Are there any locations in the well where the probability is independent of time? Explain how you determined your answer.