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- Use the standard power law expansions,

which are valid for complex
,
to prove Euler's theorem,

where
is real.
- Equations (724) and (725) can be combined with Euler's theorem to give

where
is a Dirac delta function.
Use this result to prove Fourier's theorem: that is, if

then

- A He-Ne laser emits radiation of wavelength
. How many photons are emitted per second
by a laser with a power of
? What force does such a laser exert on a body that completely absorbs
its radiation?

- The ionization energy of a hydrogen atom in its ground state is
. Calculate the
frequency (in hertz), wavelength, and wavenumber of the electromagnetic radiation that will just ionize the atom.

- The maximum energy of photoelectrons from Aluminium is
for radiation of wavelength
,
and
for radiation of wavelength
. Use this data to calculate Planck's
constant (divided by
) and the work function of Aluminium. [Adapted from Gasiorowicz 1996.]

- Show that the de Broglie wavelength of an electron accelerated across a potential difference
is given by

where
is measured in volts. [From Pain 1999.]

- If the atoms in a regular crystal are separated by
demonstrate that an accelerating
voltage of about
would be required to produce a useful electron diffraction pattern from the crystal. [Modified from Pain 1999.]

- A particle of mass
has a wavefunction

where
and
are positive real constants. For what potential
does
satisfy
Schrödinger's equation?

- Show that the wavefunction of a particle of mass
trapped in a one-dimensional square potential well of
of width
, and infinite depth, returns to its original form after a quantum revival time
.

- Show that the normalization constant for the stationary wavefunction

describing an electron trapped in a three-dimensional rectangular potential well of dimensions
,
,
, and
infinite depth, is
. Here,
,
, and
are positive integers. [From Pain 1999.]

- Derive Equation (1193).

- Consider a particle trapped in the finite potential well whose potential is given by Equation (1179).
Demonstrate that for a totally symmetric state the ratio of the probability of finding the particle outside to the
probability of finding the particle inside the well is

where
, and
. Hence, demonstrate that for a shallow well (i.e.,
)
,
whereas for a deep well (i.e.,
)
(assuming that the particle is in the ground state).

- Derive expression (1209) from Equations (1205)-(1208).

- Show that the coefficient of transmission of a particle of mass
and energy
, incident on a square
potential barrier of height
, and width
, is

where
and
. Demonstrate that the coefficient of transmission
is unity (i.e., there is no reflection from the barrier) when
, where
is positive
integer.

- The probability of a particle of mass
penetrating a distance
into a classically
forbidden region is proportional to
, where

If
and
show that
is approximately equal to
for an electron, and
for a proton. [Modified from Pain 1999.]

** Next:** Physical Constants
** Up:** Wave Mechanics
** Previous:** Square Potential Barrier
Richard Fitzpatrick
2013-04-08