Two adjacent energy levels of an electron in a harmonic potential well are known to be and . What is the spring constant of the potential well?
The spring constant for potential well=
Energy level of quantum harmonic oscillator =
denotes the state,
= reduced Planck's constant and
classical angular frequency.
Adjacent energy level=
spring constant of the potential well
Thus, the spring constant for potential well =
A particle confined in a rigid one-dimensional box of length has an energy level and an adjacent energy level .
a. Determine the values of n and n + 1.
b. Draw an energy-level diagram showing all energy levels from 1 through n + 1. Label each level and write the energy beside it.
c. Sketch the n + 1 wave function on the n + 1 energy level.
d. What is the wavelength of a photon emitted in the transition? Compare this to a typical visible-light wavelength.
e. What is the mass of the particle? Can you identify it?
Model an atom as an electron in a rigid box of length , roughly twice the Bohr radius.
a. What are the four lowest energy levels of the electron?
b. Calculate all the wavelengths that would be seen in the emission spectrum of this atom due to quantum jumps between these four energy levels. Give each wavelength a label to indicate the transition.
c. Are these wavelengths in the infrared, visible, or ultraviolet portion of the spectrum?
d. The stationary states of the Bohr hydrogen atom have negative energies. The stationary states of this model of the atom have positive energies. Is this a physically significant difference? Explain.
e. Compare this model of an atom to the Bohr hydrogen atom. In what ways are the two models similar? Other than the signs of the energy levels, in what ways are they different?
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