What physical quantity is represented by \(|\psi(x,t)|^2\) in one-dimensional quantum mechanics?

A one-dimensional wave function is normalized by \(\int |\psi|^2\,dx=1\). What are the units of \(\psi\)?

Normalize \(\psi(x)=A\sin(\pi x/L)\) on \(0\le x\le L\), with \(\psi=0\) elsewhere.

For the normalized state \(\psi=\sqrt{2/L}\sin(\pi x/L)\) on \(0\le x\le L\), find the probability of locating the particle in the left half of the box.

Normalize \(\psi(x)=Ae^{-|x|/a}\) on the whole line.

For \(\psi(x)=a^{-1/2}e^{-|x|/a}\), find the probability that \(|x|\le a\).

Explain why multiplying a wave function by a constant phase factor \(e^{i\alpha}\) does not change any position probabilities.

Let \(\psi=(\phi_1+\phi_2)/\sqrt2\), where \(\phi_1\) and \(\phi_2\) are normalized but may overlap. Expand \(|\psi|^2\) and identify the interference term.

For \(\psi=\sqrt{2/L}\sin(\pi x/L)\) in an infinite well, find \(\langle x\rangle\) without evaluating a full integral.

For \(\psi=\sqrt{2/L}\sin(\pi x/L)\) on \(0\le x\le L\), evaluate \(\langle x^2\rangle\).

Why can a nonzero constant wave function on the whole real line not describe a single localized particle?

A state is \(\psi=C(\phi_1+i\phi_2)\), where \(\phi_1\) and \(\phi_2\) are orthonormal. Find \(C\) for normalization.

At a point where \(\psi(x)=0\), what is the probability of detecting the particle exactly at that point? What is the more meaningful nearby question?

A normalized state has \(\langle x\rangle=0\) and \(\langle x^2\rangle=a^2\). Find \(\Delta x\).

A wave function is odd: \(\psi(-x)=-\psi(x)\). What can you infer about \(\langle x\rangle\) if the state is normalized on a symmetric domain?

For \(\psi=A x\) on \(0\le x\le L\), zero elsewhere, find the real positive normalization constant.

A state is \(\psi=(\phi_1e^{-iE_1t/\hbar}+\phi_2e^{-iE_2t/\hbar})/\sqrt2\), where \(\phi_1\) and \(\phi_2\) are real energy eigenfunctions. Explain when the probability density is time-dependent.

Two normalized states have overlap \(\langle\phi_1|\phi_2\rangle=s\), where \(s\) is real. Normalize \(\psi=A(\phi_1+\phi_2)\).

A normalized wave packet is squeezed so its spatial width is halved. Give a qualitative consequence for momentum uncertainty and justify it.

Prove that the probability density alone is not enough to predict all later quantum behavior, using two wave functions with the same \(|\psi|^2\) but different relative phase.