Download Lecture 8 - Pauli exclusion principle, particle in a box, Heisenberg

Lecture 8
The Pauli exclusion principle
Particle in a box, quantum pressure in solids
Inner products, Dirac notation
The Heisenberg uncertainty principle
Objectives
Learn about the Pauli exclusion principle and how it leads to the creation
of energy bands.
Learn about quantum “pressure”, a result of the Pauli exclusion principle,
and how it prevents the collapse of solids.
Learn how to construct a Heisenberg uncertainty relation for two
observables.
Bosons and fermions
Suppose we have two identical particles, e.g. electrons. The wavefunction
for a two particle system is a function of the coordinates of particle 1, x 1 ,
the coordinates of particle 2, x 2 and the time, i.e. x 1, x 2, t .
Such a wavefunction is governed by the Schrödinger equation as before
and the energy is the total energy of the system.
If particle 1 is in a state a x 1 and particle 2 in a state b x 2 then
(ignoring time) x 1, x 2 =a x 1 b x 2 , i.e. x 1, x 2 is the product of the
two wavefunctions.
However, electrons are indistinguishable from one-another in the sense
that we cannot create an experiment to tell two electrons apart, we can
put x 1, x 2 = x 2, x 1 .
We must create a wavefunction that isn’t specific as to which electron is at
which location. We write ± x 1, x 2 = A[a x 1 b x 2 ±b x 1 a x 2 ] .
Bosons and fermions II
We classify particles into two types: bosons and fermions.
Bosons: particles such as photons and mesons, these have integer spin
and use the plus sign.
Fermions: particles such as electrons and protons, these have half-integer
spin and take the minus sign.
Suppose we have two electrons with wavefunctions a and b , then
- x 1, x 2 = A[a x 1 b x 2 b x 1 a x 2 ] . If a =b then - x 1, x 2 =0 , i.e.
such a state cannot exist. This is known as the Pauli exclusion principle.
The Pauli exclusion principle applies to particles whose wavefunctions
overlap. Note that we have not considered the effect of spin. Electrons can
have two possible values of spin (+½ or -½), so the Pauli exclusion
principle in fact demands unique values of x s , where s is a
spinor describing the spin of a particle.
The Stern-Gerlach experiment
Silver atoms are emitted from a hot oven and split into two beams.
Silver has a single outer electron with zero orbital angular momentum.
The spin of the outer electron is responsible for the splitting.
The Stern-Gerlach experiment II
The silver atoms have zero angular momentum, so there should be no
interaction with the magnetic field (no moving charge = no magnetic
moment). This means that the pattern on the photographic plate should
be diffuse as each atom would have a random dipole orientation.
In fact, the pattern on the plate forms two distinct parts. This suggests
that the atoms have just two possible orientations of the magnetic
moment.
The magnetic moment arises from the “spin” of the outer electron.
We can imagine two kinds of angular momentum: orbital and spin. For
example the earth has orbital angular momentum due to its motion
around the sun. It also has spin angular momentum due to its rotation
about the north-south axis.
In quantum mechanics, spin is an intrinsic property of a particle, like mass
or charge.
Band structure
As atoms become closer together the wavefunctions of the electrons begin
to overlap. The Pauli exclusion principle results in a smearing of the
allowed electron energies into bands. For some elements band gaps are
formed, electrons cannot have energies in the band gap region.
Energy bands vs. atomic separation, r, in semiconductors
Free electron gas
In the solid state, in particular in metals, the most loosely bound electrons
are free to move around. The electron gas theory of Sommerfeld considers
these electrons as free particles in a box.
We start by assuming the electrons are confined to a box of size lx, ly and
lz. They are free to move inside the box but cannot escape it. This is
similar to the infinite square well model, except that we have three
dimensions instead of one.
The potential acting on the electron is V x , y , z=0 for 0 xl x , 0 yl y
and 0 zl z and V x , y , z= otherwise.
2
2
The Schrödinger equation is = E , we can separate the
2m
wavefunction into x, y and z components x , y , z= X x Y y Z z .
2
2
2
d2 X
d2Y
d2 Z
=Ex X ,
=E y Y
=E z Z
So
and
with
2 m dx 2
2 m dy 2
2 m dz 2
E = E x
E y
E z .
Solutions to the Schrödinger equation
Let k x
2 m Ex
=
, ky
2m Ey
=
and k z
2 m Ez
=
2
d X
2
=k
so that
x X etc.
2
dx
X x = Ax sin k x x
B x cos k x x ,
The
general
solutions
are
Y y = A y sin k y y
B y cos k y y and Z z= Az sin k z z
B z cos k z z .
X 0=Y 0=Z 0=0
and
We
know
that
at
the
boundaries
X l x =Y l y =Z l z =0 so B x = B y = B z=0 . Therefore k x l x =n x , k y l y =n y and k z l z =n z , where nx, ny and nz are integers.
l
2
2 nx x dx=1
To find the normalisation coefficients we must solve Ax sin
lx
0
2
2 lx
=1 and Ax =
So Ax
etc. The wavefunction for the electron becomes
lx
2
x
n n
x
y
nx ny nz 8
sin
x sin
y sin
z .
n x , y , z=
l xl y lz
lx
ly
lz
z
k-space
The energy of the wavefunction is given by
2
2
2
2 2 n x n y n z 2 k 2
where k is the magnitude of the wave
En ,n ,n =
2
2 =
2
2m lx ly lz
2m
vector k = k x , k y , k z . Each k vector can be associated with a volume of kspace
x
y
z
The volume associated with each k vector
3
3
=
.
is
lx l yl z V
Because of the Pauli exclusion principle
only two electrons (spin-up and spindown) can occupy any given state.
Quantum pressure
Suppose an object contains N atoms, each of which contributes q free
electrons. The k-space will be filled, starting with the lowest energy
states, and will have the shape of an octant of a sphere (because kx, ky and
kz are positive).
3
Nq 1 4
The volume of the k-space is given by
(the number of
k 3f =
8 3
2 V
states times the volume of each state), kf is the radius of the sphere.
Nq
2 1/3
, then k f =3 .
V
The surface of the sphere of k-space is called the Fermi surface. The
2 2
k f 2
energy of the Fermi surface is E f =
=
3 2 2 / 3 .
2m 2 m
We define the free electron density as =
Quantum pressure II
We can calculate the total energy of the electron gas by considering the
energy of a spherical shell of thickness dk. The volume of the shell is
2
k
dk
1
2
, and the number of states in the shell is
V k = 4 k dk =
8
2
2 2
2V k
Vk 2 dk
k
nk = 3
=
. Each state has an energy
, so the energy of the
/V 2
2m
2 2
2
k Vk dk
dE=
.
The
total
energy
is
then
shell
is
2 m 2
k
2 5
2
2
5/3
2
V
k
3
Nq
V
f
2/ 3
4
=
V
k
dk
=
E=
.
2
2
2
2m 0
10 m 10 m f
If the box expands by an amount dV then the energy changes by an
2
2
5 /3
2 3 Nq 2 dV
5/ 3
dE=
V
dV
=
E
amount
, i.e. the energy decreases.
2
3
3 V
10 m Conversely, if the box shrinks the energy increases.
Quantum pressure III
We can conceive of a quantum pressure which acts on the Fermi surface
and prevents the box from collapsing.
If we try to shrink the size of the box the work done is dW = PdV , so
2
2
5/3
3
Nq
2 E 2
5/ 3
P= = V
, the quantum pressure resists the shrinking
3 V 3
10 m 2
of the box.
The quantum pressure is sometimes called the degeneracy pressure and is
a result of the requirement that the electrons must have different
wavefunctions.
The quantum pressure is one reason why a cold solid object cannot
collapse.
Quantum pressure prevents the collapse of cold objects
Images of ultracold Li atom clouds. Li-7 is a boson and Li-6 is a fermion.
The Pauli exclusion principle leads to a “pressure” which prevents the
fermion cloud from shrinking. The same effect prevents white dwarf stars
from collapsing under their own gravity.
Inner products (scalar product)
We saw before that vectors can be written as | . We can also consider
functions in a similar way. Dirac introduced a bracket notation to
represent the scalar product of two vectors or functions.
*
*
*
In Dirac notation we can write ==a1 b1 a 2 b 2 ...
an bn .
Suppose | =1
i x and | =1i y , then =0 , i.e. the two vectors
are orthogonal. Also =1
i1i=2, = , which is the norm
(length) of the vector.
For a set of mutually orthogonal, normalised vectors we can write
i j = ij , where ij =1 for i= j and ij =0 for i j .
For two functions f x and g x the inner product can be written as
f g = f x * g x dx . For this to work properly we must set the condition
that
f x2 dx .
Inner products II
In Dirac notation the inner product of two functions is written as 1 2 .
The symbol | 2 is known as a ket whilst the symbol 1 | is known as a
bra.
From the definition we can write 1 2 = 21 *
If c is a complex number we have
1c 2 =c 1 2 *
c 1 2 =c 1 2 3 1
2 = 3 1 3 2 If the two functions are orthogonal then 1 2 =0
If the wavefunctions are normalised then 1 1 = 2 2 =1
There are some transformations T such that T = T , these are
known as Hermitian transformations.
Heisenberg uncertainty principle
We can write the expectation values of x and p in Dirac notation as
x =x and p = p .
The deviation of a measurement of x or p from x or p is x = x x and p = p p . The corresponding operators are x x and p p .
2
2
So x = x x and p = p p .
2
2
Both of these operators are Hermitian so x = x x x x 2
p = p p p p .
2
2
Let x= x x and p= p p . Then x = xx and p = p p .
2
2
and
2
2
The Schwartz inequality says x p = xx p p x p .
For
2
a
complex
2
2
number
2
z = Re z Im z Im z .
z,
Im z=
1
z z * 2i
and
Heisenberg uncertainty principle
2
1
2 2
x p px .
So, if z= x p we can write x p 2i
It can be shown that x p= x p x p and px = p x x p .
So x p px = x p p x =[ x , p ] , where [ x , p ] is the commutator of the
two operators.
2
1
2 2
[ x , p ] .
So 2 p 2i
We know that x = x and p =i , we can use a test function, f(x), to
work out [ x , p ] .
d
d
[ x , p ] f x= x i f i xf
dx
dx
df
df
[ x , p ] f x=i x
i x f =i f x and so [ x , p ]=i .
dx
dx
2
2
1
Therefore and x p .
i =
2
2i
2
The same argument holds for other incompatible observables.
2
x
2
p
Conclusions
The Pauli exclusion principle states that the wavefunctions of fermions in
the same state cannot overlap. This leads to the creation of energy bands
in solids and a quantum “pressure” which prevents the collapse of solid
materials.
The Heisenberg uncertainty principle can be determined for any two
incompatible observables from the corresponding operators.