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- Calculate the Clebsch-Gordon coefficients for adding spin one-half
to spin one.

- An electron in a hydrogen atom occupies the combined spin
and position state whose spinor-wavefunction is

Here,
are the eigenstates of
corresponding to the eigenvalues
, respectively,
and
,
,
are conventional spherical coordinates.
- What values would a measurement of
yield, and with
what probabilities?
- Same for
.
- Same for
.
- Same for
.
- Same for
.
- Same for
.
- What is the probability density for finding the electron at
,
,
?
- What is the probability density for finding the electron in the
spin-up state (with respect to the
-axis) at radius
?

[61]

- In a low energy neutron-proton system (with zero orbital angular
momentum) the potential energy is given by

where
is the vector connecting the two particles,
,
**
**
denotes the vector of the Pauli matrices of the neutron,
and
**
**
denotes the vector of the Pauli matrices of the proton. Calculate
the potential energy for the neutron-proton system:
- In the spin singlet (i.e., spin zero) state.
- In the spin triplet (i.e., spin one) state.

[Hint: Calculate the expectation value of
with respect to the overall spin state.] [53]

- Consider two electrons in a spin singlet (i.e., spin zero) state.
- If a measurement of the spin of one of the electrons shows that it
is in the state with
, what is the probability that a
measurement of the
-component of the spin of the
other electron yields
?
- If a measurement of the spin of one of the electrons shows
that it is in the state with
, what is the probability that a
measurement of the
-component of the spin of the
other electron yields
?
- Finally, if electron 1 is in a spin state described by
, and
electron 2 is in a spin state described by
, what is
the probability that the two-electron spin state is a triplet (i.e., spin one) state?
Here,
are the eigenstates of
corresponding to the eigenvalues,
, respectively, for
the electron in question. [53]

** Next:** Time-Independent Perturbation Theory
** Up:** Addition of Angular Momentum
** Previous:** Calculation of Clebsch-Gordon Coefficients
Richard Fitzpatrick
2016-01-22