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- According to the
*Larmor formula* of classical physics, a non-relativistic electron whose instantaneous acceleration is of magnitude
radiates
electromagnetic energy at the rate

where
is the magnitude of the electron charge,
the permittivity of the vacuum,
and
the velocity of light in vacuum [49]. Consider a classical electron in a
circular orbit of radius
around a proton. Demonstrate that the radiated energy would cause the orbital radius to decrease in time according to

where
is the Bohr radius,
the
electron mass,
the reduced Planck constant, and

Here,
is the fine structure constant.
Deduce that the classical lifetime of a ground-state electron in a hydrogen atom is
.

- Let the
, for
, be a set of
*orthonormal* kets that span an
-dimensional ket space.
By orthonormal, we mean that the kets are mutually orthogonal, and have unit norms, so that

for
. Show that

- Demonstrate that

in a finite-dimensional ket space.

- Demonstrate that in a finite-dimensional ket space:

Here,
,
,
are general operators.

- If
,
are Hermitian operators then demonstrate that
is only Hermitian provided
and
commute. In addition, show that
is Hermitian, where
is a positive integer. [53]

- Let
be a general operator. Show that
,
, and
are Hermitian operators. [53]

- Let
be an Hermitian operator. Demonstrate that the Hermitian conjugate of the operator
is
. [53]

- Suppose that
and
are two commuting operators. Demonstrate that

- Let the
be the normalized eigenkets of an observable
, whose corresponding eigenvalues,
, are discrete.
Demonstrate that

where the sum is over all eigenvalues.

- Let the
, where
, and
, be a set of degenerate unnormalized eigenkets of some
observable
. Suppose that the
are not mutually orthogonal. Demonstrate that a set of
mutually orthogonal (but unnormalized) degenerate eigenkets,
, for
, can be constructed as follows:

This process is known as *Gram-Schmidt orthogonalization*.

- Demonstrate that the expectation value of an Hermitian operator is a real number. Show that the
expectation value of an anti-hermitian operator is an imaginary number.

- Let
be an Hermitian operator. Demonstrate that
.

- Consider an Hermitian operator,
, that has the property that
. What are the eigenvalues of
? What are the eigenvalues if
is not restricted to
being Hermitian? [53]

- An operator
is said to be
*unitary* if

Show that if
and
then
. [53]

- Show that if
is Hermitian then
is unitary. [53]

- Show that if the
, for
, form a complete orthonormal set, so that

then the
, for
, where
is unitary, are also orthonormal. [53]

- The eigenstates of some operator
acting in an
-dimensional ket space are written

for
, where all of the
are real. Suppose that the
are orthonormal, and
span the ket space. Deduce that
is Hermitian.

- Let
be an observable whose eigenvalues,
, lie in a continuous range. Let the
, where

be the corresponding eigenkets. Demonstrate that

where the integral is over the whole range of eigenvalues.

- Show that

where
is a Dirac delta function, and
a constant.

** Next:** Position and Momentum
** Up:** Fundamental Concepts
** Previous:** Continuous Spectra
Richard Fitzpatrick
2016-01-22