1
Review of Phys 300
The Classical Point of View
A system is a collection of particles that interacts themselves via internal
forces and that may interact with the world outside via external fields.
Intrinsic properties of a classical system (e.g. rest mass and
charge) are independent of its physical environment and
therefore don’t depend on the particle’s location and don’t
evolve with time.
Extrinsic properties of a classical system (e.g. position and
momentum) evolve with time in response to the forces on the
particle.
According to the classical physics, all intrinsic and extrinsic properties of
a particle could be known to infinite precision and we could measure the
precise value of both position and momentum of a particle at the same
time (classical physics describes a determinate universe).
Classical physics predicts the trajectory (i.e., the values of its position and
momentum for all times after some initial time 0t ) of a particle,
0, ;r t p t t t trajectory , where the linear momentum is given by
vd
p t m r t m tdt
with m the mass of the particle. Trajectories are
called the state descriptors of Newtonian physics. The evolution of the
state represented by the trajectory is given by 2
2,
dm r t V r t
dt where
,V r t is the potential energy of the particle. To obtain the trajectory for
0t t , one only need to know ,V r t and the initial conditions ( the values
of r and p at the initial time 0t ).
As a result, classical physics (Newton’s laws) predicts the future of any
system that is known its initial conditions.
2
Near the end of the 19th century, theoretical physics was based on three
fundamentals:
Newton’s theory of mechanics
Maxwell’s theory of electromagnetic phenomena
Thermodynamics and kinetic theory of gasses
At the end of the 19th century and the beginning of the 20th century, a crisis
appeared in physics. A series of experimental observations couldn’t be
explained by concepts of classical physics.
Blackbody radiation
Photoelectric effect
Compton effect
Particle diffraction
The repeated contradiction of classical laws made it necessary to develop
some new concepts and the development of these concepts emerged
“Quantum Theory” that introduced new concepts:
Blackbody radiation
behaviour of radiation Photoelectric effect
Compton effect
behaviour o
f
Particle diffraction
Particle
Wave
Quantization of physical quantities
particle
Electrons at t
he atomic states mwit L nh vr
Quantum physics is a theory describing the properties of matter at the level
of micro phenomena (molecule, atom, nucleus, elementary particle, …).
This theory provides the answers to many questions which remained
unsolved in classical physics.
3
The Quantum Point of View
The concept of a particle doesn’t exist in the quantum world, particles
behave both as a particle and a wave (wave-particle duality).
Unlike the classical physics, nature itself will not allow position and
momentum to be resolved to infinite precision (Heisenberg uncertainty
relation), 0 02
xx t p t where 0x t is the minimum uncertainty in
the measuremen of the position in the x-direction at time 0t and 0xp t is
the minimum uncertainty in the measurement of the momentum in the x-
direction at time 0t . Position and momentum are fundamentally
incompatible observables (the universe is inherently uncertain).
Therefore, if we cannot know the position and momentum of a particle at
0t , we cannot specify the initial conditions of the particle and hence
cannot calculate the trajectory.
Physical quantities (like energy) take some certain values (quantization).
4
Heisenberg uncertainty relations
The standard deviation or uncertainty for two operators A and B are given
by
22 22 22 2ΔA = = 1
ΔA ΔB ,2
A A A AA B
iA A
The quantity 22ΔA= A A measures the spread of values about the
mean for A.
Wave-particle duality
Particles behave both as a particle and a wave in the quantum world
Matter
WavePhoton
Wave
MatterMatter
Physical quantities
for particles
Physical quantities
for waves
E h
p /h
What is the matter wave?
Plane wave in 1D: wt)i(kxe At)ψ(x,
dt
dx
tΔ
xΔlimv
0tΔp
constantwtkx “phase”
v2
1
mv
mv)2/1(
p
h
h
Eνλ
/λ2π
ν 2π
k
w
dt
dxv
2
p
speed particle
speed Broglie de
p v 2
1 v plane wave cannot represent a particle.
5
Wave packet: the superposition of the waves that is approximately
localized in space at any given time).
constantt2
dwx
2
dk v
mdv
mvdv
dp/
dE/
dk
dw
dt
dxvg
speed particle
speed Broglie de
g v v
wave packet represents a particle.
The wave packet is a superposition of waves.
Superposition of waves Fourier integrate
/1( , ) (p)
2
i px Etx t dp e
that is a solution of a partial differential equation called Schrodinger
Equation a free particle of mass m
2
22
x
t)ψ(x,
2m-
t
t)ψ(x, i
Schrodinger Equation in the presence of a potential V(x) will be
t)ψ(x,)x(Vx
t)ψ(x,
2m-
t
t)ψ(x, i
2
22
where (x,t) is the solution of Schrodinger equation, mathematical
description of the wave packet. It is to be physically acceptable solution
(called wave function) if it satisfies square integrable (finite), single-valued
and continuous properties. Any solution of Schrodinger equation that
becomes infinite must be discarded.
5 10 15 20 25 30 35
2
1
0
1
2
6
The Schrödinger equation has two important properties
The equation is linear and homogeneous. An important consequence
of this property is that the superposition principle holds. This means
that if 1ψ (x,t) and
2ψ (x,t) are solutions of the Schrödinger equation,
then the linear combination of these functions is also a solution
( , ) ( , )i ii
x t C x t .
The equation is first order with respect to time derivative (meaning
that the state of a system at some initial time to determines its
behavior for all future times)
2 2
2
( , ) ( , ) -
2first order in time derivative
x t x ti
t m x
It has one initial condition. Thus, if ( , 0)x t is known, ( , )x t can be
found. Firstly, (p) is calculated from - /1( ) ( ,0)
2
ipxp dx x e
. Then,
using (p) , t)ψ(x, is obtained from i(px-Et)/1ψ(x,t) dp (p) e
2 π .
Every solution of Schrodinger equation is not a wave function (physically
acceptable solution). To be an acceptable solution, an eigenfunction
( , )x t and its derivative ( , ) /x t x are required to have the following
properties:
( , )x t and ( , ) /x t x must be continuous.
( , )x t and ( , ) /x t x must be single valued.
( , )x t and ( , ) /x t x must be finite (square integrable).
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Probability interpretation of the wave function:
Finite property requires the interpretation of probability density. t)ψ(x, is in
general a complex function and is apparently not a measurable quantity.
However, the wave function is a very useful tool for calculating other quite
meaningful, mathematically real quantities. 2|t)ψ(x,| is always real. It is
large where the particle is supposed to be, and small elsewhere. We also
know that 2|t)ψ(x,| is spreading with time (it means that as time passes, it is
less probable to find the particle where it is put at t=0). 2|t)ψ(x,|t)P(x, is
called the probably density and requires:
-
2 1dx|t)ψ(x,| that is called the
normalization condition.
Conservation of Probability
The probability density is defined as , * , ,x t x t x t . We know
that ,x t satisfies the Schrödinger equation. It is also true that * ,x t
satisfies the Schrödinger equation. It satisfies the following equation
called continuity equation
, ( , ) 0x t j x tt x
where *
*2
d dj
mi dx dx
is the flux (probability current density that
is the number of particles per second passing any point x )
In 3D, they will be
, ( , ) 0r t j r tt
and ( , ) * *2
j r tmi
We then find
s as 0t),x(jsdt),x(j xdt) P(x,xdt
S-
3
-
3
constantt)x P(x,d -
3
A change of the density in a region bxa is equal to a net change in the
flux into that region ( , ) ( , ) ( , ) ( , )b b
a a
ddx P x t dx j x t j a t j b t
dt x
8
Expectation value:
How can we calculate the measurable quantities (position, momentum,
energy, etc) from the wave function?
An average value of a measurement (statistical average of a large number
of measurements) is called the expectation value.
The expectation value of the position of a particle:
In general, x-coordinate of a particle cannot have a certain value in a
region if 0t)ψ(x, in that region. It is possible to talk about the average
value of x-coordinate. It is called the expectation value of x-coordinate
and is given by
-
*
-
dx t)ψ(x,x t)(x,ψdx t)x P(x,x
The expectation value of the momentum of a particle:
-
* dx t)ψ(x, xi
t)(x,ψp
where xi
p
is the momentum operator in x-space.
Alternatively, in momentum space
*
-
x dp (p) i (p) p
and *
-
p dp (p) p (p)
where p
ix
is the position operator in momentum space.
For an arbitrary function, the expectation value
-
* dx t)ψ(x, f(x) t)(x,ψx)(f and
-
* dx t)ψ(x, xi
f t)(x,ψp)(f
Example:
-
2*2 dx t)ψ(x, x t)(x,ψx and
-
2
*2 dx t)ψ(x, xi
t)(x,ψp
The uncertainties in x and p
22 xxΔx and 22 ppΔp
The quantity 22 xxΔx measures the spread of values about the
mean for x and 22 ppΔp for p .
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Hermitian Operator
The operator xi
p
is complex, but its expectation value is real. Such
operators (position, momentum, energy, etc) are called Hermition
operators.
The energy operator : t
iE
Hamiltonian of the system: )x(Vxm2
H2
22
Operators play a central role in Quantum Mechanics. Products of
operators need careful definition, because the order in which they act is
important.
AB-BAB,A is called a commutation relation between operators
Commutation relation between momentum and position: ixp,
This relation is independent of what wave function this acts on. This is a
fundamental commutation relation in quantum physics. Heisenberg
uncertainty relation becomes
2
ΔpΔx x,pi4
1ΔpΔx
222
Important concepts:
The notation of a wave packet as representing a particle
Schrodinger equation as fundamental equation in quantum physics
The wave function t)ψ(x, that has the probability interpretation
Heisenberg uncertainty relation
Statistical average of a large number of measurements as expectation
value
10
Eigenvalues and Eigenfunctions
Time-dependent Schrodinger Equation
t)ψ(x,)x(Vx
t)ψ(x,
2m-
t
t)ψ(x, i
2
22
The method of separation of variables: if the spatial behavior of the wave
function does not change with time, we use some time-varying multiplying
factor in front of the spatial part of the wave function.
Eu(x)V(x)u(x)dx
u(x)d
2m-
CeT(t)ET(t)dt
dT(t)i
)x(u)t(Tt)ψ(x,
2
22
iEt/-
The eigenvalue equation
ˆ O ( )ˆ ˆ O O
ˆ O f(x) λ f(x) eigenvalueoperator
eigenfunction eigenfunctionof wrt f x
of of
The eigenvalue equation states that the operator Q acting on certain
functions f(x) (eigenfunction) will give back these functions multiplied by
a constant λ (eigenvalue).
Energy eigenvalue equation
(x)u E (x)uH EE
Although the wave function -iEt/
E e)x(ut)ψ(x, depends on time, the
probability density 2
E
2 |(x)u||t)ψ(x,| does not depend on time.
The initial spatial wave function as a linear superposition of the energy
eigenstates
ψ(x,0) ( )n n
n
C u x
The evolution of the wave function in time is given by a simple linear
superposition of these eigenstates
n
t/-iE
nnne)x(uCt)ψ(x,
11
Alternatively and equivalently, a time-evolution operator - /iH te is applied
to an initial state0ψ(x,t ) at time
0t to obtain the evaluated state 1ψ(x,t ) at time
1t as
1 0- /
1 0( , ) ( , )iH t t
x t e x t
Example : a particle in an infinite box
0x ,
ax0 ,0
0x ,
V(x)
BcoskxAsinkxu(x)0u(x)mE2
dx
u(x)d22
2
Energy eigenfunction and eigenvalues
1,2,3,...n 2ma
nπE
a
xn πsin
a
2(x)u
2
222
n
n
Orthonormality condition
nm when1
nm when0(x)u(x)udx mn
a
0
m
*
n δ
The expansion postulate: Any function ψ(x) can be expanded in terms of
orthogonal functions.
(x)uAψ(x) n
n
n
where a
0
*
nn ψ(x)(x)udx A is the projection of ψ(x) onto nu (x) .
The expectation value of energy is given by
n
n
2
n E |A|ψ(x)|H|ψ(x)
where 2
n |A| is the probability that a measurement of the energy for the state
ψ(x) yields the eigenvalue nE .
12
Momentum eigenfunction
ipx/
pppope e π 2
1(x)u (x)u p (x)u p
Degeneracy: if more than one eigenfunction corresponds to the same
eigenvalue, this eigenvalue is said to be degenerate.
Parity
1λλψ(x)ψ(-x)ψ(x)P
parity offunction eigen also ision eigenfunctEnergy
motion ofconstant a is P0H,P
Important concepts:
The eigenvalue equation
Particle in a box problem
The expansion postulate
13
One-Dimensional Potentials
The Potential Step
a) When a neutron with an external kinetic energy K enters a nucleus, it
experiences a potential
b) While a charged particle moves along the axis of two cylindrical
electrodes held at different voltages, its potential energy changes very
rapidly when passing from one to the other. Potential energy function
can be approximated by a step potential.
The Potential Well
a) The motion of a neutron in a nucleus can be approximated by assuming
that the particle is in a square well potential with a depth about 50 MeV.
b) A square well potential results from superimposing the potential acting
on a conducting electron in a metal.
14
The Potential Barrier
a) scattering problems
b) Emission of α particles from radioactive nuclei
c) Fusion process
d) Tunnel diode
e) Cold emission electrons
The Harmonic Oscillator
15
The General Structure of Wave Mechanics
Postulates
Postulate 1:
The dynamical state of a particle can be described by a wave function
which contains all the information that can be known about the particle. Postulate 2:
An arbitrary function can be expressed as a linear superposition of a set of
orthonormal functions.
Postulate 3:
The Schrodinger equation describes the behavior of the wave function in
space and time. Postulate 4:
Each observable quantity q can be directly associated with a linear,
Hermitian operator. The value q is an eigenvalue of the operator.
Hermitian property: 2121 ψQψQψψ
Theorem 1: The eigenvalues of a Hermitian operator are real.
Theorem 2: The eigenfunctions of a Hermitian operator are orthogonal
if they correspond to distinct eigenvalues. Postulate 5:
The expectation value of a measurement of a variable q is given
mathematically as
ψψ
ψQ|ψq
Each observable quantity q can be directly associated with a linear,
Hermitian operator. The value q is an eigenvalue of the operator.
Commuting Observables:
The cummutator of two operators A and B is defined by AB-BAB ,A .
If the commutator vanishes when acting on any wave function, the
operators A and B are said to commute, ABBA .
When the operators commute, 0B ,A , their observables A and B are said
to be compatible. Observables are non-compatible if 0B ,A .
If two observables are compatible, their corresponding operators have the
simultaneous eigenfunctions and A and B are said to be simultaneously
measurable. Thus, compatible observables can be measured simultaneously
with arbitrary accuracy, non-compatible observables cannot.
16
nnnn ψBaψABψBA 0B ,A
nψB is an eigenfunction of A belonging to the eigenvalue na . Since na is
non-degenerate, nψB can only differ from nψ by a multiplicative constant
which we can call nb
nnn ψbψB
Thus we see that nψ is simultaneously an eigenfunction of the operators A
and B belonging to the eigenvalues na and nb , respectively.
If . . . ,C ,B ,A are a set of commuting operators, there is a simultaneous
eigenfunction nψ of . . . ,C ,B ,A with the eigenvalues ,...c ,b ,a nnn .
Time Dependence and Classical Limit:
The expectation value of an operator A is
-
*
tdx t)ψ(x, A t)(x,ψA
The expectation value varies with time as
t
tt
A,Hi
t
AA
dt
d
If A has no explicit time dependence, tt
A,Hi
Adt
d
. The observable A is
a constant of motion if the operator A commutes with H.
17
Operator Methods in Quantum Mechanics
One dimensional harmonic oscillator Hamiltonian is
222
xmω2
1
2m
pH where ixp,
The problem is how to find the energy eigenvalues and eigenstates of this
Hamiltonian.
1. Polynomial Method
2 2
1/2
α x /2
n nn
n
αu (x) e H (α x)
π 2 n!
1 E n ω n 0,1,2,....
2
2. Operator Method
Dimensionless position and momentum operators are defined as
q-ip
mω
1p
x mω
q
x
Two non-hermitian operators are introduces in terms of q and p as
p mω
1ix
mω
2
1ip)(q
2
1A
p mω
1ix
mω
2
1ip)(q
2
1A
x
x
Since x and p are Hermition, A is indeed the hermitian conjugate of A .
Commutator A,A gives 1AAAA1A,A .
The Hamiltonian in terms of these operators becomes
2
1AAH ω
where AAN is known as the number operator which is Hermitian.
0NH, They have simultaneous eigenfunctions.
n|nn|N
n|En|H n
18
The other commutation relations become
AωAHHAAωAH,
AωAHHAAωAH,
Consider n|En|H n . Multiply both sides with A as
n|Aω)E(n|AHn|EAn|HA nn
AωAH
n|A is an eigenfunction of H with the eigenvalue ω)E( n such that the
energy nE is lowered by one unit of ω .
Multiply again with A
n|A ω)2E(n|A Hn|Aω)E(An|AHA 2n
2n
AωAH
n|A2 is also an eigenfunction of H with the eigenvalue ω)2E( n such that
the energy nE is lowered by ω2 .
The operator A is called a lowering operator.
Since the harmonic oscillator has only positive energy states including zero,
there must be a lower bound on the energy. There is a state of lowest
energy, the ground state.
00|A
So that energy cannot be lowered any more.
ωωω
ω
2
1E0|
2
10|
2
10|AA
0|2
1AA0|H
0
Now multiply n|En|H n with A
n|Aω)E(n|AHn|EAn|HA nn
AωAH
n|A is an eigenfunction of H with the eigenvalue ω)E( n such that the
energy nE is raised by one unit of ω .
Multiply again with A
n|A ω)2E(n|A Hn|Aω)E(An|AHA2
n
2
n
AωAH
n|A2 is also an eigenfunction of H with the eigenvalue ω)2E( n such that
the energy nE is raised by ω2 .
19
The operator A is called a raising operator.
We obtained the energy spectrum of the harmonic oscillator without solving
any differential equation as
0,1,2,....n ω2
1n En
What are the values of nC and nD ?
1n|Dn|A
1n|Cn|A
n
n
Using AA,N and AA,N , we get 1nCn and nDn .
The eigenstates are
0|A!n
1n|
.
.
.
0|A1.2.3
12|A
3
13| :2n
0|A1.2
11|A
2
12| :1n
0|A1
11| :0n
n|A1n
11n|1n|1nn|A
n
3
2
The explicit for of the eigenstates are
2/qnnn
2/q0
n
n
2/q000
0
0
2
2
2
e)q(HN)q(u
eCq
q2
1
!n
1u|
eCu0qudq
ud
dq
dip ,0u|)ipq(
2
1
00|A
20
The time dependence of operators:
The solution of time dependent Schroidnger equation is ψ(0)|eψ(t)| /iHt
The expectation value of an operator B is
0
/iHt/iHt
t)t(Bψ(0)|)t(B| ψ(0)ψ(0)|eBe| ψ(0)ψ(t)|B| ψ(t)B
Pictures
Schrodinger picture:
Operators are time-independent.
Time evaluation of the system is determined by a time dependent wave
function.
Heisenberg picture:
Operators are time-dependent.
Wave functions are time-independent.
The result is the same whatever picture we use. Time variation of )t(B is
given by
S t ω i
H
S tω i
H
Ae(t)A
A e(t)A(t)BH,
i(t)B
dt
d
As an example: position and momentum operators of a particle
tx(0)sin ω mω- tp(0)cosωp(t)
tsinωmω
p(0) tx(0)cos ωx(t)
21
Levi-Civita Tensor
Coordinate transformation in 3D can be written as
1,2,3i xλx3
1j
jij'i
The magnitude of a vector is invariant under coordinate transformation
(rotation)
'rr
'z'y'x'r
zyxr
222
222
or
22'rr
3 32 2 ' '
i i ij j ik ij ik j
i=1 jk 1
ij ik
' x x x λ x λ x λ λ x x
λ λ "orthogonality condition"
i i k k
i i j k j
jk
j
r r x
ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
ij
ijk k
i i j j i j i j i i
i j ij i
i i j j i j i j i k ijk k
i j ij ije
i ijk i j ijk mk i jm im j
jk k
A B A e B e A B e e A B
C A B A e B e A B e e A B e
C A B
1 if i, j, k form an even permutation of 1, 2, 3
1 if i, j, k form an odd permutation of 1, 2, 3
0 when any two indices are the same
ijkε
x
y
'z,z
,z,b
b
'x
θ
θ
z
y
x
1 0 0
0 1 θ-
0 θ 1
z'
y'
x'
z
y
x
1 0 0
0 cosθ sinθ-
0 sinθ cosθ
z'
y'
x'rotation
malinfinitesi
'y
z'
y'
x'
1 0 0
0 cosθ sinθ
0 sinθ- cosθ
z
y
x
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