Doubly ionized lithium , , absorbs a photon and jumps from the ground state to its n=2 level. What was the wavelength of the photon?

Consider a particle in the ground state of a finite well. Describe the changes in its wave function and energy as the walls are made progressively higher (U₀ is increased) until essentially infinite.

Question: Consider an electron in the ground state of a hydrogen atom. (a) Calculate the expectation value of its potential energy. (b) What is the expectation value of its kinetic energy? (Hint: What is the expectation value of the total energy?)

It is shown in section 6.1 that for the E<U₀ potential step, . Use it to calculate the probability density to the left of the step:

1. Show that the result is, where . Because the reflected wave is of the same amplitude as the incident, this is a typical standing wave pattern varying between 0 and .
2. Determine and D in the limits k and tend to 0 and interpret your results.

If things really do have a dual wave-particle nature, then if the wave spreads, the probability of finding the particle should spread proportionally, independent of the degree of spreading, mass, speed, and even Planck’s constant. Imagine that a beam of particles of mass m and speed v, moving in the x direction, passes through a single slit of width w . Show that the angle at which the first diffraction minimum would be found ( , from physical optics) is proportional to the angle at which the particle would likely be deflected , and that the proportionality factor is a pure number, independent of m, v, w and h . (Assume small angles: ).

A harmonic oscillator has its minimum possible energy, what is the probability of finding it in the classically forbidden region? (Note: At some point, a calculator able to do numerical integration will be needed.)