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1. Review of Basic Quantum Mechanics
Contents
Review quantum mechanics postulates, angular momentum theory and hydrogen atom
electronic structure.
Notes
Review1: Postulates of Quantum Mechanics
Postulate 1. The state of a quantum mechanical system is completely
specified by a function
that depends on the coordinates of the
particle(s) and on time. This function, called the wave function or
state function, has the important property that
the probability that the particle lies in the volume element
at at time .
is
located
The wavefunction must satisfy certain mathematical conditions because of
this probabilistic interpretation. For the case of a single particle, the probability
of finding it somewhere is 1, so that we have the normalization condition
(1.1)
It is customary to also normalize many-particle wavefunctions to 1. The
wavefunction must also be single-valued, continuous, and finite.
Postulate 2. In any measurement of the observable associated with
operator
,the only values that will ever be observed are the
eigenvalues , which satisfy the eigenvalue equation
(1.2)
This postulate captures the central point of quantum mechanics--the values of
dynamical variables can be quantized (although it is still possible to have a continuum
of eigenvalues in the case of unbound states). If the system is in an eigenstate of
with eigenvalue , then any measurement of the quantity
will yield .
Although measurements must always yield an eigenvalue, the state does not
have to be an eigenstate of
. An arbitrary state can be expanded in the
complete set of eigenvectors of
(
as
(1.3)
where
may go to infinity. In this case we only know that the measurement of
will yield one of the values
, but we don't know which one. However, we
do know the probability that eigenvalue
squared of the coefficient,
will occur--it is the absolute value
. This leads to our third postulate.
Postulate 3. If a system is in a state described by a normalized wave
function
,then the average value of the observable corresponding to
is given by
(1.4)
Postulate 4. To every observable in classical mechanics there
corresponds a linear, Hermitian operator in quantum mechanics.
This postulate comes about because if we require that the expectation value of an
operator
is real, then
must be a Hermitian operator. Some common operators
occuring in quantum mechanics are collected in Table 1:
Table 1: Physical observables and their corresponding quantum operators (single particle)
Observable
Observable
Operator Operator
Name
Symbol
Symbol
Position
Operation
Multiply by
Momentum
Kinetic energy
Potential energy
Multiply by
Total energy
Angular momentum
Postulate 5. The wavefunction or state function of a system evolves in
time according to the time-dependent Schrödinger equation
(1.5)
The central equation of quantum mechanics must be accepted as a postulate.
Postulate 6. The total wavefunction must be antisymmetric with
respect to the interchange of all coordinates of one fermion with those
of another. Electronic spin must be included in this set of coordinates.
[To be studied in the next lecture].
Review 2: The Angular Momentum Operators
We start from the classical expression for angular momentum,
to obtain the quantum mechanical version
, where
,
,
, and
are all three-dimensional vectors. This definition leads immediately to
expressions for the three components of L:
=
(2.1)
=
(2.2)
=
(2.3)
From these definitions, we may easily derive the following commutators
=
(2.4)
=
(2.5)
=
(2.6)
where the indices i,j,k can be x, y, or z, and where the coefficient
is unity if i,j,k
form a cyclic permutation of x,y,z[i.e., (x,y,z), (y,z,x), or (z,x,y)] and -1 for a reverse
cyclic permutation [(z,y,x), (x,z,y), or (y,x,z)]. The final commutator indicates that we
cannot generally know Lx, Ly, and Lzsimultaneously except if we have an eigenstate
with eigenvalue 0 for each of these.
Classically, any component of the angular momentum must be less than or
equal to the magnitude of the overall angular momentum vector. Quantum
mechanically, the average value of any component of the angular momentum
must be less than or equal to the square root of the expectation value of
dotted with itself:
(2.7)
is simply
Since
.
, we can have simultaneous
commutes with any component
eigenfunctions of
and a given component
since the expression for
. We usually pick the z axis,
is the easiest of the three when we work in
spherical polar coordinates:
=
(2.8)
=
(2.9)
=
(2.10)
Of course it is also possible to express
in terms of the unit vectors for
spherical polar coordinates,
=
(2.11)
=
(2.12)
=
(2.13)
Here,
(2.14)
and
=
=
=
(2.15)
The simultaneous eigenfunctions of
harmonics,
and
are called the spherical
, where lis the total angular momentum quantum
number, and m is the so-called magnetic quantum number. The spherical
harmonics are defined as
(2.16)
where Plm are the associated Legendre polynomials. We require that
and spherical harmonics with m<0 are defined in terms of the spherical harmonics
with m>0 according to Ylm = (-1)m [ Yl-m]*. The spherical harmonics are normalized
over integration of angular coordinates such that
,
(2.17)
and they have the following special properties:
=
(2.18)
=
(2.19)
It can be useful to define ladder operators for angular momentum. The
following ladder operators work not only for straight angular momentum
but also for combined angular momenta such as
,
. If
(2.20)
then
(2.21)
We can see that these ladder operators raise or lower the magnetic quantum number m
but leave l alone.
One can also show that in spherical polar coordinates
(2.22)
By comparing this expression with that for
in spherical polar coordinates,
(2.23)
we can see that the Hamiltonian can be written as
=
=
Clearly
(2.24)
commutes with the kinetic energy term,
has no r dependence.
Likewise, if
, then
commutes with the whole Hamiltonian. Hence,
for problems where the potential depends only on r (central force problems), we can
find simultaneous eigenfunctions of
,
, and
.
The most important property for an angular momentum operator is
eq.(2.6) from which almost all other properties can be derived. For the
electron, the spin is just a form of angular momentum, but it is intrinsic
meaning it cannot be expressed into the form like eqs.(2.1-3) or
eqs.(2.15,22). However, we can still find out the eigenvalues of the
spin angular momentum operators of the electron. [To be studied in
the next lecture].
Review 3. Hydrogen Atom
Finally, consider the hydrogen atom as a proton fixed at the origin, orbited by an
electron of reduced mass
. The potential due to electrostatic attraction is
(3.1)
in SI units. The kinetic energy term in the Hamiltonian is
(3.2)
so we write out the Schrödinger equation in spherical polar coordinates as
(3.3)
It happens that we can factor
into
again the spherical harmonics. The radial part
, where
are
then can be shown to obey the
equation
(3.4)
which is called the radial equation for the hydrogen atom. Its (messy) solutions are
(3.5)
where
, and
is the Bohr radius,
. The functions
are the associated Laguerre functions. The hydrogen atom
eigenvalues are
(3.6)
There are relatively few other interesting problems that can be solved
analytically. For molecular systems, one must resort to approximate solutions.
When the electron spin is considered, to a good approximation, the
Hamiltonian of hydrogen atom will be the same if there is no magnetic field
applied to it. Therefore the energy eigenequation will be the same except the
eigenfunctions are modified so that the spin wavefunctions are taken into
account:
ψ
nlm(r,θ
φ)→ψ
nlm(r,θ
φ ) g(ms)
(3.7)
where g(ms) is the spin state which is generally a linear combination of
α and β: g(ms)=c1α+c2β. There are two choices for c1 and c2. Therefore,
the degeneracy of hydrogen atom eigenstate is changed into 2n2.
[To be studied in the next lecture].
References
Textbook pp.1-282.
Assignment
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