Quantum Formalism
In the previous chapters we have focused on the some simple quantum mechanical systems, namely, infinite well
potential, finite well potential, delta function, free particle and harmonic oscillator potential. We have obtained
solution of the Schrodinger equation for given potentials. It is very important note that, we have discussed
properties of the wave function. Now, in this chapter we will summarize those properties.
Space of the Wavefunction: Hilbert Space
Note that wave function live in the Hilbert space. In this space the functions will be square integrable such
that:
∞
∫ 𝜓 ∗ 𝜓𝑑𝑥 < ∞
−∞
In the hilbert space the wave function has vector properties. We define the vectors:
∞
⟨𝑓|𝑔⟩ = ∫ 𝑓 ∗ 𝑔𝑑𝑥
−∞
where < 𝑓| is calledbra and |𝑔 > is called ket vector. The notation is known ans Bra-ket or Dirac notation.
These vectors satisfy the Schwarz inequality:
|⟨𝑓|𝑔⟩| ≤ √⟨𝑓|𝑓⟩⟨𝑔|𝑔⟩
As an example of application of the notation, consider the vectors
𝑎1
𝑏1
|𝛼 > = (𝑎2 ) ; |𝛽 > = (𝑏2 )
𝑎3
𝑏3
then
⟨𝛼|𝛽⟩ = 𝑎1∗ 𝑏1 + 𝑎2∗ 𝑏2 + 𝑎3∗ 𝑏3
or
⟨𝛽|𝛼⟩ = 𝑏1∗ 𝑎1 + 𝑏2∗ 𝑎2 + 𝑏3∗ 𝑎3
Properties of the Observables
Expectation values of the observable can be written in the Dirac notation
⟨𝑂⟩ = ⟨𝜓|𝑂|𝜓⟩
A measurable operator should be a Hermitian operator. Because hermitian operators real eigenvalues. Hermitian
operator is defined as:
⟨𝑂⟩ = ⟨𝑂⟩∗
An observable is represented by a Hermitian operator.
Another important properties of the operators can be described as follows. If the state function (wave function)
eigen function of the operator O then the expectation values and eigenvalues of the operator is equal to each
others and the state is known as determined state. (see lecture notes)
Collection of all eigenvalues of an operator is called spectrum.
Sometimes two or more linearly independent wave function have the same eigenvalues. In this case the
spectrum is said to be degenerate.
If the spectrum of an operator is discrete then the wave function is normalizable.
If the spectrum of the operator is continuous then wave function cannot be normalized but the other
properties still hold. In this case one can use fourier transformation. (See lecture notes free particle)
Uncertainty Principle
Without further discussion (see text book for its proof) uncertainity between two operator is given by
Δ𝐴Δ𝐵 ≥ |
1
⟨[𝐴, 𝐵]⟩|
2𝑖
The simple interpretation is if two operator commute then the corresponding observables can be measured
separately without any uncertainty.
The whole of quantum mechanics can be expressed in terms of a small set of postulates. This chapter introduces
the postulates and illustrates how they are used.
Postulates of Quantum Mechanics
Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics.
Quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical
realm. Quantum mechanics is a very powerful theory which has led to an accurate description of
the micro-physical mechanisms. It is founded on a set of postulates from which the main processes pertaining to
its application domain are derived. A challenging issue in physics is therefore to exhibit the underlying principles
from which these postulates might emerge.
The set of statements we find in the literature as ‘postulates’ or ‘principles’ can be split into three subsets: the
main postulates which cannot be derived from more fundamental ones, the secondary postulates which are often
presented as ‘postulates’ but can actually be derived from the main ones, and then statements often called
‘principles’ which are well known to be as mere consequences of the postulates.
Main postulates
(1) State or Wavefunction. Each physical system is described by a state function which determines all can be
known about the system. The coordinate realization of this state function, the wavefunction Ψ(𝑟, 𝑡) (uppercase
psi) plays a central role in quantum mechanics. Two wavefunctions represent the same state if they differ only by
a phase factor. The wavefunction has to be finite and single valued throughout position space, and furthermore, it
must also be a continuous and continuously differentiable function. We shall see that the wavefunction of a
system will be specified by a set of labels called quantum numbers, and may then be written ca,b, . . . , where a,
b, . . . are the quantum numbers. The values of these quantum numbers specify the wavefunction and thus allow
the values of various physical observables to be calculated.
(2) Equation for the wave function (Schrödinger equation). The time evolution of the wavefunction of a nonrelativistic physical system is given by the time-dependent Schrödinger equation
𝑖ℏ
𝜕Ψ
= 𝐻Ψ
𝜕𝑡
ℏ2 𝜕2
where the Hamiltonian 𝐻 = −
+ 𝑉(𝑥) is a linear Hermitian operator, whose expression is constructed
2𝑚 𝜕𝑥 2
from the correspondence principle. The time independent form of the equation can be written as:
𝐸𝜓 = 𝐻𝜓
We shall have a great deal to say about the Schrödinger equation and its solutions in the rest of the text.
(3) Correspondence principle. To every dynamical variable of classical mechanics there corresponds in quantum
mechanics a linear, Hermitian operator. When the operators act upon the wavefunction associated with a definite
value of that observable yields this value times the wavefunction. Mathematically we can write:
𝑂𝜓 = 𝑜𝜓
Where O is operator, o is eigenvalue of the operator O and 𝜓 is eigenfunction. The equation is named
eigenvalue equation. The requirement that the operators are hermitian ensures that the observables
have real values. Selection of the operators for all observables depends on the position and momentum operator.
The more common operators occurring in quantum mechanics for a single particle are listed below and are
constructed using the position and momentum operators.
Operator
Abreviation
1D
Math representation
3D
1D
3D
Position
x
r
Momentum
𝑝𝑥
𝑝
K, T, KE
K, T, KE
Hamiltonian (time dependent)
E, H
E, H
Hamiltonian (time independent)
E,H
E,H
𝑝𝑥2
+ 𝑉(𝑥)
2𝑚
𝑝2
+ 𝑉(𝑟)
2𝑚
𝐿𝑥
𝐿
𝑦𝑝𝑧 − 𝑧𝑝𝑦
𝒓×𝒑
Kinetic Energy
Angular momentum
x
−𝑖ℏ
r
𝑑
𝑑𝑥
𝑝𝑥2
2𝑚
𝑖ℏ
𝜕
𝜕𝑡
−𝑖ℏ∇
𝑝2
2𝑚
𝑖ℏ
𝜕
𝜕𝑡
(4) Measurements (Von Neumann’s postulate). If a measurement of the observable A yields some value 𝑎𝑖 , the
wavefunction of the system just after the measurement is the corresponding eigenstate 𝜓𝑖 . The expectation value
will simply be the eigenvalue 𝑎𝑖 .
(5) Born’s postulate: probabilistic interpretation of the wavefunction. The squared norm of the wavefunction
|𝜓|2 𝑑𝜏 is interpreted as the probability of the finding of the particle in volume 𝑑𝜏. More specifically, if 𝜓(𝑟, 𝑡) is
the wavefunction of a single particle, 𝜓 ∗ (𝑟 , 𝑡)𝜓(𝑟, 𝑡)𝑑𝜏 is the probability that the particle lies in the volume
element 𝑑𝜏 located at 𝜏 at time t. Because of this interpretation and since the probability of finding a single
particle somewhere is 1, the wavefunction of this particle must fulfil the normalization condition
∞
∫ 𝜓 ∗ (𝑟 , 𝑡)𝜓(𝑟, 𝑡)𝑑𝜏 = 1
−∞
Secondary postulates
One can find in the some books other statements which are often presented as ‘postulates’ but
which are mere consequences of the above five ‘main’ postulates. We examine below some of them and show
how we can derive them from these ‘main’ postulates.
(1) Superposition principle. Quantum superposition is the application of the superposition principle to quantum
mechanics. It states that a linear combination of state functions of a given physical system is a state function of
this system. This principle follows from the linearity of the H operator in the Schrödinger equation, which is
therefore a linear second order differential equation to which this principle applies.
(2) Eigenvalues and eigenfunctions. Any measurement of an observable A will give as a result one of the
eigenvalues a of the associated operator A, which satisfy the equation Aψ = aψ. This is the consequences of
corresponding principle.
(3) Expectation value. For a system described by a normalized wavefunction ψ, the expectation value of an
observable A is given by
∞
⟨𝐴⟩ = ∫ 𝜓 ∗ 𝐴𝜓𝑑𝑥
−∞
This statement follows from the probabilistic interpretation attached to ψ, i.e., from Born’s Postulate.
(4) Expansion in eigenfunctions. The set of eigenfunctions of an operator A forms a complete set of linearly
independent functions. Therefore, an arbitrary state ψ can be expanded in the complete set of eigenfunctions of A
(𝐴𝜓𝑛 = 𝑎𝑛 𝜓𝑛 ), i.e., as
𝜓 = ∑ 𝑐𝑛 𝜓𝑛
𝑛
where the sum may go to infinity. For the case where the eigenvalue spectrum is discrete and non-degenerate and
where the system is in the normalized state ψ, the probability of obtaining as a result of a measurement of A the
eigenvalue 𝑎𝑛 is |𝑐𝑛 |2 . This statement can be straightforwardly generalized to the degenerate and continuous
spectrum cases. The statements
∞
1 𝑖𝑓 𝑛 = 𝑚
∫ 𝜓𝑛∗ 𝜓𝑚 𝑑𝑥 = {
0 𝑖𝑓 𝑛 ≠ 𝑚
−∞
Hermitian operators are known to exhibit the two following properties: (i) two eigenvectors of a Hermitian
operator corresponding to two different eigenvalues are orthogonal, (ii) in an Hilbert space with finite dimension
N, a Hermitian operator always possesses N eigenvectors that are linearly independent. This implies that, in such
a finite dimensional space, it is always possible to construct a base with the eigenvectors of a Hermitian operator
and to expand any wavefunction in this base.
(5) Probability conservation. The probability conservation is a consequence of the Hermitian property of ˆH.
This property first implies that the norm of the state function is time independent and it also implies a local
probability conservation which can be written
𝜕𝜌 𝜕𝑗
+
=0
𝜕𝑡 𝜕𝑥
Where 𝜌 = |𝜓|2 𝑎𝑛𝑑 𝑗(𝑥, 𝑡) =
ℏ
2𝑖𝑚
(
𝜕Ψ
𝜕𝑥
Ψ∗ − Ψ
𝜕Ψ∗
𝜕𝑥
).
(6) Reduction of the wave packet or projection hypothesis. This statement does not need to be postulated since
it can be deduced from other postulates. It is actually implicitly contained in von Neumann’s postulate. In
quantum mechanics, wave function collapse (also called collapse of the state vector or reduction of the wave
packet) is the process by which a wave function—initially in a superposition of different eigenstates—appears to
reduce to a single one of the states after interaction with an observer. In simplified terms, it is the condensation
of physical possibilities into a single occurrence, as seen by an observer.
Derived principles.
(1) Heisenberg’s uncertainty principle. If p and x are two conjugate observables such that
They are not commute, it is easy to show that they satisfy the relation
Δ𝑥Δ𝑝 ≥
ℏ
2
In general
Δ𝐴Δ𝐵 ≥
1
|⟨[𝐴, 𝐵]⟩|
2
where the root mean square deviation of A is defined as
Δ𝐴 = √⟨𝐴2 ⟩ − ⟨𝐴⟩2
Finally, it is appropriate at this point to make a few remarks about the so-called energy–time uncertainty relation.
Although the energy operator is well defined (it is the hamiltonian for the system), there is no operator for time
in quantum mechanics. Time is a parameter, not an observable. Therefore, strictly speaking, there is no
uncertainty relation between energy and time. The true significance of the energy–time ‘uncertainty principle’ is
that it is a relation between the uncertainty in the energy of a system that has a finite lifetime 𝜏 (tau), and is of
the form 𝛿𝐸 = ℏ/2𝜏.
(2) The spin-statistic theorem. The spin-statistics theorem states that integer spin particles are bosons, while
half-integer spin particles are fermions. The theorem states that:
The wave function of a system of identical integer-spin particles has the same value when the positions of any
two particles are swapped. Particles with wavefunctions symmetric under exchange are called bosons;
The wave function of a system of identical half-integer spin particles changes sign when two particles are
swapped. Particles with wavefunctions anti-symmetric under exchange are called fermions.
(3) The Pauli exclusion principle. Two identical fermions cannot be in the same quantum
state. This is a mere consequence of the spin-statistic theorem.
Example 1. Show that any linear combination of the complex functions 𝑒 2𝑖𝑥 𝑎𝑛𝑑 𝑒 −2𝑖𝑥 is an eigenfunction of the
operator
𝑑2
𝑑𝑥 2
, where 𝑖 2 = −1.
Solution. Then we operate on a linear combination of the functions.
𝑑2
(𝑎𝑒 2𝑖𝑥 + 𝑏𝑒 −2𝑖𝑥 ) = −4(𝑎𝑒 2𝑖𝑥 + 𝑏𝑒 −2𝑖𝑥 )
𝑑𝑥 2
The linear combination satisfies the eigenvalue equation and has the same eigenvalue (-4) as do the two complex
functions.
Self-test 1. Show that any linear combination of the functions 𝑠𝑖𝑛4𝑥 𝑎𝑛𝑑 𝑐𝑜𝑠4𝑥 is an eigenfunction of the
operator
𝑑2
𝑑𝑥 2
.
Example 2. An operator A has eigenfunctions 𝜓1 , 𝜓2 , . . . , 𝜓𝑛 with corresponding eigenvalues 𝑎1 , 𝑎2 , . . . , 𝑎𝑛 . The
state of a system is described by a normalized wavefunction 𝜓 given by
𝜓=
1
3
3
𝜓 − √ 𝜓2 + √ 𝜓3
2 1
8
8
What will be the outcome of measuring the observable A?
Method. First, we need to determine if c is an eigenfunction of the operator A. If it is, then we shall obtain the
same eigenvalue of A in every measurement. If it is not, we shall obtain different values in a series of different
measurements. In the latter case, if we have an expression for c in terms of the eigenfunctions of A, then we can
determine what different values are possible, the probabilities of obtaining them, and the average value from a
large series of measurements.
Solution. To test whether 𝜓 is an eigenfunction of the operator A we proceed as follows:
1
3
3
𝐴𝜓 = 𝐴 [ 𝜓1 − √ 𝜓2 + √ 𝜓3 ]
2
8
8
1
3
3
= 𝑎1 𝜓1 − √ 𝑎2 𝜓2 + √ 𝑎3 𝜓3 ≠ 𝑎𝜓
2
8
8
Therefore, 𝜓 is not an eigenfunction of A. However, because 𝜓 is a linear combination of 𝜓1 , 𝜓2 𝑎𝑛𝑑 , 𝜓3 we
will obtain, in different measurements, the values 𝑎1 , 𝑎2 , 𝑎𝑛𝑑 𝑎3 (the eigenvalues of the eigenfunctions of A that
1 3
3
contribute to 𝜓). The probabilities of obtaining 𝑎1 , 𝑎2 , 𝑎𝑛𝑑 𝑎3 are, respectively, , 𝑎𝑛𝑑 .
4 8
8
The average value, given by
⟨𝐴⟩ =
1
3
3
𝑎1 + 𝑎2 + 𝑎3
4
8
8
Self-test 2. Repeat the problem using
𝜓=
1
7
1
𝜓 − √ 𝜓2 − 𝑖 𝜓3
3 1
9
3
References
Laurent Nottale and Marie-Noelle Celerier, J. Phys. A: Math. Theor. 40 (2007) 14471–14498
Atkins & Friedman: Molecular Quantum Mechanics