Verify that $A \sin (k x)+B \cos (k x)$ is a solution of equation we saw earlier.

The term interaction is sometimes used interchangeably with force, and other times interchangeably with potential energy. Although force and potential energy certainly aren't the same thing, what justification is there for using the same term to cover both?

A tiny $1 \mu g$ particle is in a 1 cm wide enclosure and takes a year to bounce from one-end to the other and back, (a) How many nodes are there in its enclosure? (b) How would your answer change if the particle were more massive or moving faster?

What is the probability that a particle in the first excited (n=2) state of an infinite well would be found in the middle third of the well? How does this compare with the classical expectation? Why?

Where would a particle in the first excited state (first above ground) of an infinite well mostly likely be found?

Because protons and neutrons are similar in mass, size, and certain other characteristics, a collective term, nucleons, has been coined that encompasses both of these constituents of the atomic nucleus. In many nuclei, nucleons are confined to dimensions of roughly 15 femtometers. Photons emitted by nuclei as the nucleons drop to lower energy levels are known as gamma particles. Their energies arc typically in the MeV range. Why does this make sense?

An electron is trapped in a quantum well (practically infinite). If the lowest-energy transition is to produce a photon of 450 nm wavelength, what should be the well's width?

An electron in the n = 4 state of a 5 nm wide infinite well makes a transition to the ground state, giving off energy in the form of a photon. What is the photon's wavelength?

Write out the total wave function $\Psi(x, t)$ for an electron in the n = 3 state of a 10 nm wide infinite well, Other than the symbols x and t, the function should include only numerical values.

A study of classical waves tells us that a standing wave can be expressed as a sum of two traveling waves, Quantum-mechanical traveling waves, are of the form $\Psi(x, t)=A e^{i(k x-\omega t)}$ Show that the infinite well's standing-wave function can be expressed as a sum of two traveling waves.

A classical particle confined to the positive x-axis experiences a force whose potential energy is $U(x)=\frac{1}{x^{2}}-\frac{2}{x}+1 \quad$ (SI units) (a) By finding its minimum value and determining its behaviors at x = 0 and $x=+\infty,$ sketch this potential energy. (b) Suppose the particle has an energy of 0.5 J. Find any turning points. Would the particle be bound?(c) Suppose the particle has an energy of 2.0 J. Find any turning points. Would the particle be bound?

A comet in an extremely elliptical orbit about a star has, of course, a maximum orbit radius. By comparison, its minimum orbit radius may be nearly 0. Make plots of the potential energy and a plausible total energy E versus radius on the same set of axes. Identify the classical turning points on your plot.

A student of classical physics says, "A charged particle, like an electron orbiting in a simple atom, shouldn't have only certain stable energies; in fact, it should lose energy by electromagnetic radiation until the atom collapses." Answer these two complaints qualitatively, appealing to as few fundamental claims of quantum mechanics as possible.

Under what circumstance does the integral $\int_{x_{0}}^{\infty} x^{b} d x$ diverge? Use this to argue that a physically acceptable wave function must fall to 0 faster than $|x|^{-\frac{1}{2}}$ does as x gets large.