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- According to 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
electron mass. 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 reduced Planck constant, and

Here,
is the velocity of light in a vacuum, and
the fine structure constant.
Deduce that the classical lifetime of a hydrogen atom is
.

- 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.

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

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

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

where the sum is over all eigenvalues, and
denotes the unity operator.

- Let the
, where
, and
, be a set of degenerate 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 a 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
, where
is the unity
operator. What are the eigenvalues of
? What are the eigenvalues if
is not restricted to
being 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, and
denotes the unity operator.

** Next:** Position and Momentum
** Up:** Fundamental Concepts
** Previous:** Continuous Spectra
Richard Fitzpatrick
2013-04-08