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Quantum Mechanics 2: Principles of Quantum Mechanics: Lecture notes and Qs
Physics (0FHH0032)
University of Hertfordshire
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Quantum Mechanics 2: Principles of Quantum Mechanics
2 Introduction
Let’s summarise some of the theoretical concepts introduced in Lecture Notes 1. Each
quantum state can be represented by a ket that lies in an abstract vector space V.
There are states for which there is no uncertainty in the value of the observable that will
be measured. The discrete set of allowed energy states
n
E for a particle in a box is one
example. They form a basis for the vector space, so that any generic energy state can
be expressed by the linear superposition
nn
n
aE
. The complex valued
coefficients
nn
aE are the probability amplitudes (the projection of state onto
the basis state
n
E
). The modulus squared
2
n
a gives the probability that a measurement
of energy will return the (real) value n
E
.
Generalising the notation, we can express a general state vector as a linear
combination of basis vectors n , that is sufficiently large (complete), so that:
n
n
an
. (2)
Valid basis vectors form an orthogonal and normal set. This means that for any two
vectors mn, in the set,
mn
mn where the Konecker delta
1,
0,
mn
mn
mn
.
Example:
In the Cartesian coordinate system, the basis vectors are
1 2 3
ˆ ˆ ˆ e e e, , i j k, ,. A general
vector r is expressed as 1 2 3
ˆ
kk
k
a a a a
r e i j k , where
ˆ
kk
a er. The basis vectors
are complete, orthogonal and normal. If
1 2 3
s bi bj bk
, is another vector, the dot
product
ii
i
ab
rs is a real number.
In a vector space with basis vectors i , a general state is expressed as
i
i
ai
,
where
i
ai .
As basis states are orthogonal and normal, the inner product of state
i
i
ai
with
state
i
j
bj
is defined by
ii
i
ba
. This definition implies that:
ii
i
ab
. (2)
To see why these formal ideas are useful we next introduce the quantum mechanical
concept that each observable (any physical quantity such as position, momentum, energy
etc) is represented by a linear operator that can act on a quantum state (a ket) to
provide the possible values of a measurement of that observable in that state.
2 Operators
A linear operator on a vector space is an object
ˆ
Q that transforms kets into kets in a
linear way (for example
ˆ
Q ). Given complex numbers ab, and kets , , a
linear transformation is defined by:
ˆ ˆ ˆ
Q a b aQ bQ. (2)
We start by introducing some important linear operators.
The projection operator
If is a normalised ket, the projection operator
ˆ
P picks out the portion of
any other ket that “lies along” , so for example:
ˆ
P . (2)
Remember that
is a just a (complex) number, so can be moved as shown.
ˆ
P
projects onto the one dimensional subspace spanned by .
Projection operators are termed idempotent, that is:
2
ˆˆ
PP.
Note that
i
P is the probability that a measurement of energy on a system in the state
will return the value
i
E. Therefore
ˆ
ii
i
H PE
represents an average value for
the energy of the state .
Example:
A particle is confined in a potential box so that its allowed energy levels are
2
n 1
E n E,
where n1, 2, 3,.... The corresponding energy states are
1 2 3
E ,E ,E ,..., or more
succinctly, 1 , 2 , 3 ,....,....
At time t 0 the particle is in the superposition state:
4
0 0 10 2 0 3 a 4.
(a) If the energy is measured at time t 0 , what is the probability of measuring the
energy
4
E?
(b) If the energy is measured at time t 0 , what is the probability of measuring an energy
value smaller than
1
6 E?
(c) What is the expectation value of energy at time t 0?
To answer these probability questions about states we must ensure that the states are
properly normalised. The normalisation condition for a state is (see Lecture 1):
2
1
i
i
a
.
Normalisation can be checked using the orthonormal property of the basis states:
1
0
nm
nm
nm
nm
.
In our case normalisation requires,
2 2 2 2
4
0 0 0 0 0 a 1.
Part (a):..
2
44
Pa0.
Part (b):..
22
12
PP 0 0 0.
Part (c):..
1 1 2 2 3 3 4 4
2 2 2
1 2 3 4
2 2 2
2 2 2 2
1
1
= 0 0 0 0.
0 1 0 2 0 3 0 4
=13.
E P E P E P E P E
E E E E
E
E
.
This result seems reasonable. The expectation value (average value) lies between the
minimum energy
1
E and the maximum energy
1
16 E.
2 Eigenvalue equations
Linear operators are generally associated with a set of eigenvectors and eigenvalues.
Consider the Hamiltonian operator
ˆ
H. If we apply
ˆ
H to the basis eigenvector
k
E , and
use the orthonormality of the eigenstates
i
E we get:
ˆ
k i i i k k k
i
H E E E E E E E
. (2)
Equation (2):
ˆ
k k k
H E E E
is an eigenvalue equation. The special determinate
state
k
E is an eigenvector. The real number
k
E
is the corresponding (measured)
eigenvalue. Learn to recognize the format of an eigenvalue equation:
ˆ
Operator eigenvector (real) eigenvalue eigenvector.
We will see shortly that the energy eigenvalue equation
ˆ
k k k
H E E E is just the time
independent Schrodinger equation.
There are corresponding eigenvalue equations for other operators. The eigenvalues of an
operator give the possible results of a measurement of the observable represented by that
operator. As measurement outcomes are real numbers the operators of quantum
mechanics must have real eigenvalues. Such operators are termed Hermitian. Let
, be two eigenstates of a linear operator
ˆ
Q, then
ˆ
Q is Hermitian if:
ˆˆ
QQ . (2)
By analogy with the discrete orthonormal relation
jk
jk , the corresponding relation
for the kets x is:
x x x x
, where
xx
is the Dirac delta function. (2)
Two useful properties of the delta function are:
x x 0 if x x, and dx x x 1
. (2)
The analogue of the discrete normalisation condition
2
1
i
i
a
becomes:
2
dx x 1
. (2)
This normalisation condition allows the following familiar physical interpretation of the
wave function
x (due to Born):
The probability to find a particle in a given interval
ab, is (see figure below):
2
,
b
a
P a b x dx
. (2)
As we will work a lot using the position representation it is worth comparing the
general operator notation with the position representation. Examples tabulated below.
Normalisation
2
1
i
i
a
2
dx x 1
Identity operator ˆ
i
I i i
ˆ
I dx x x
General operator and
position operator
ˆ
i i i
i
Q q q q
ˆ x dxx x x
Inner (scalar) product of
kets, and
ii
i
ba
.
dx x x
dx x x
Expectation value ˆ
QQ.
ˆ
Q x Q x dx
.
Observable Operator (1-D form), Eigenvalue equation
Position
ˆ x x xˆ x x.
Momentum
piˆ
x
.
ˆ
p p p p.
Potential Energy
ˆ
VV x.
Kinetic Energy
22
2
ˆ
ˆ
2
p
Ti
mx
Total Energy
22
2
ˆ
()
2
ˆ ˆ ˆ
H V x
mx
H T V
.
ˆ
H E E E.
This is the time independent
Schrodinger eqn.
Example:
The Hamiltonian operator that appears in the energy eigenvalue equation corresponds to
the classical measure of total energy. Consider the one-dimensional motion of a particle.
Classically its energy is:
2
2
1
22
x
p
E T V mx V x V x
m
.
The corresponding quantum operator is the Hamiltonian:
2
ˆ
ˆ ˆ ˆ ˆ
2
x
p
H T V V x
m
.
With reference to the operator table above we see that:
ˆ ˆ
AB,0
. (2)
Writing this out in full we get:
ˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆ
A B,0 AB BA A B B A
.
The brackets in the final terms emphasise that it is important to respect the order of
operations acting on the ket
(which are from right to left). The following are some
useful rules for manipulating commutators:
, , ,
,
, , ,
A B C A C B C
AB BA A B
AB C A C B A B C
. (2)
Example:
Let two operators
ˆ ˆ
ABand with respective eigenvalues and
nn
ab share the common
basis of eigenkets n , so that:
ˆ ˆ
and
nn
A n a n B n b n.
Let the commutator
ˆ ˆ
AB,
act on the general state vector,
n
n
cn
. Then:
ˆ ˆ ˆˆ ˆˆ ˆˆ ˆˆ
,
ˆˆˆˆ
= 0
nn
nn
n n n n n n n n n n n
n n n
A B c AB BA n c AB n BA n
c Ab n Ba n c b A n a B n c b a n a b n
.
Thus
ˆ ˆ
AB,0
if
ˆ ˆ
AB, commute.
Example:
Consider the 1-D position and momentum operators
ˆˆ ,
x
x p i
x
. Then, using the
shorthand notation for
xx , we have:
ˆ ˆ, ˆˆ ˆ ˆ
0
x x x
x p xp p x
x i i x i x i i x i
x x x x
.
Note the need to use the product rule of differentiation in the second line.
We could write this result in shorthand as
ˆˆ
,
x
x p i. Note also that
ˆˆ
,
x
p x i.
We see that the position and momentum operators do not commute. This means that
they do not share a common basis of eigenkets. As a result there will be a mutual
uncertainty between position and momentum measurements. This is a statement of the
Heisenberg Uncertainty Principle (see below).
Note also that
ˆ ˆ ˆ ˆ ,,
yz
y p z p i
. Check for yourself that:
ˆ ˆ ˆ ˆ
, , 0
ˆ ˆ ˆ ˆ
, , 0
ˆˆˆˆ
, , 0
yz
xz
xy
x p x p
y p y p
z p z p
.
Example:
We now show that the Hamiltonian operator and the linear momentum operator of a free
particle (V 0 ) do commute, so energy and momentum can be measured compatibly.
We use the same shorthand as above. From our table of operators we have:
2 22
2
22
2
3 2 2 3 3 3
2 2 3 3
ˆ
ˆ
, and ˆ ,
22
ˆ
,ˆ ( ) , ( )
2
( ) ( )
( ) 0
22
x
x
x
p
H p i
m m x x
H p x i x
m x x
xx
i x i
m x x x x m x x
.
Note that Hamiltonian operator for a free particle is just the kinetic energy operator.
2 The General Uncertainty Principle.
Consider a system in a state . We make a series of measurements of two observables
AB, represented by the operators
ˆ ˆ
AB,
, ensuring the system is always in the same state
before each measurement of AB or is made. If the resulting (standard deviations) in
the measurements are,
11
22
2222
a a a , b b b.
Consider the specific transition
21
EE10 in atomic hydrogen. The principle
wavelength is 121 nm while the excited state lifetime is τ = 1 ns, so:
7
2 10 eV
2
E
.
As
2
hc hc
E hE
.
Using Eabove, gives
15
2 10 m
. This wavelength spread represents the
natural linewidth of the photon emitted in the transition. Such a small value gets masked
by the wavelength shifts resulting from the thermal motion of the emitting atoms or by
velocity shifts due collisions between atoms.
2 Basic principles of Quantum Mechanics
In this lecture we have presented several theoretical concepts. It is useful to summarise
the relation between the physical world and these abstract concepts in a set of quantum
principles. Various authors will summarise the key principles differently. Below is one
summary of concepts.
Principle 1
The state of a system is completely described by a state function. A general state is
typically denoted by a ket . States are described as ‘vectors’ belonging to a vector
space (Hilbert space). A general state can be represented as a linear superposition
i
i
ai
in some basis i. In the position basis x , the wave function (a
probability amplitude) is written as
xx.
The relation between states and the physical world depends on the following two
principles.
Principle 2
To every physical quantity Q there corresponds a linear operator
ˆ
Q. The operators for
position and momentum
ˆˆ
xp, , satisfy the commutation relation,
ˆˆ x p, i. This non-
zero value means that the observables ,
x
xp cannot be measured simultaneously to
arbitrary accuracy. The measurements must satisfy the Uncertainty principle.
Principle 3
The eigenvalues
i
qof an operator
ˆ
Q
are the possible results of measuring the physical
quantity Q. They correspond to the definite eigenstates
i
qi. The eigenvalues and
eigenstates are obtained from the eigenvalue equation
ˆ
i
Q i q i. For a normalised
generic state
i
i
ai
, the probability of measuring the eigenvalue
i
q is
2
i
a where
i
ai and
2
1
n
n
a
.
The above principles allow the treatment of single particles in static potentials. Two more
principles are needed to treat time evolution of states and many-particle systems. For
reference, they are:
Principle 4
There are only two kinds of particles in nature: bosons (described by symmetric state
kets) and fermions (described by antisymmetric state kets).
Principle 5
The time evolution of a system (between measurements) is determined by the first-order
linear equation, the time dependent Schrodinger equation:
ˆ
i t H t
t
.
We will be working with this equation in the next lecture.
Recommended Reading
Binney & Skinner: Chapter 2
Rae: Chapter 4
Davies & Betts: Chapter 6
Auletta et al: Chapter 2
Quantum Mechanics 2: Principles of Quantum Mechanics: Lecture notes and Qs
Module: Physics (0FHH0032)
University: University of Hertfordshire
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