The wave function Ψ provides information about a quantum mechanical system. It is used to calculate the probability of finding a particle in a particular region of space. For example, if a ball is constrained to move in one dimension within a tube of length L, the wave function Ψ would be constant, with a value determined by the normalization condition. Integrating this wave function over the left or right half of the tube then gives the 50% probability of finding the ball in each half. Operators are used in quantum mechanics to extract measurable properties like position, momentum, or energy from the wave function.

1. Wave Function :
It is an essential element of a quantum mechanical system by using it we can get any
meaningful information about the system. The symbol for wave function in quantum
mechanics is Ψ(written as “psi” and pronounced as “sigh”.

2. Presenter :Asad Ali
Bsf1800659
Dr. Masood

3. Examples:
 Lets say we are dealing in classical mechanics with a mass spring system.
Hook's Law F = -kx (1)
Newton’s Law F = ma (2)
F = m
𝑑𝑥2
𝑑𝑡2
Comparing (1) and (2),we get,
m
𝑑𝑥2
𝑑𝑡2
+ kx = 0
After solving we get a result,
x(t) = A(cos ωt-Φ)
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4. Construction of a wave
function:
 Suppose a ball is constrained to move along a line inside a tube of length L. The ball is
equally likely to be found anywhere in the tube at some time t. What is the probability of
finding the ball in the left half of the tube at that time(The answer is 50%,but how do we
get this answer by using the probabilistic interpretation of quantum mechanical wave
function?
Strategy:
The first step is to write down the wave function. The ball is equally like to be found
anywhere in the box, so one way to describe the ball with a constant wave function.The
normalization condition can be used to find the value of function and a simple integration over
half of box yields the final answer.

5. Solution:
 The wave function of the ball can be written as
Ψ(x,t) = C (0 < x < L) ,where C is a constant
 We can determine the value of constant C by applying the normalization condition(we set
t=0 to simplify the notation).

6. Examples of Operators:

7. Origin of Operators in Quantum
Mechanics
 Operators were introduced by Dirac but without any mathematical justification,then Von
Neumann introduced the required mathematical description for operators. Operators in
quantum mechanics are used to extract information about any measurable parameter from a
given wave function.
 Quantum mechanics tells us that every object has a wave nature associated with it. This
wave nature is more prominent in particles at the atomic and subatomic level and hence
their dynamics may be explained by the quantum theory. This also means that every
particle at the quantum scale has a wave function associated with it which contains
information regarding several measurable parameters. These parameters may be the total
energy of a particle, its momentum, its position at a particular instant of time or its angular
momentum. In order to extract particular information, we require a particular operator.
<f(x)> = 𝟎
∞
𝜳 ∗ 𝒙, 𝒕 𝒇 𝒙 𝜳 𝒙, 𝒕 𝒅𝒙
<P> = 𝟎
∞
𝜳 ∗ 𝒙, 𝒕 𝑷 𝜳 𝒙, 𝒕 𝒅𝒙

8.  In quantum mechanics, the simultaneous measurement of P and x is not possible so P can’t
be expressed as function of x and t. At this point there was need of operators in quantum
mechanics for Momentum and Energy Expectation value problems.
 Consider a quantum particle which is moving along x axis in free space. The wave function
for this particle is;
Ψ(x,t) = cos(kx-ωt) + i sin(kx-ωt)

9. 𝝏Ψ(x,t)
𝝏𝒙
= -k sin (kx-ωt) + 𝒊 kcos(kx-ωt)
𝝏Ψ(x,t)
𝝏𝒙
= 𝒊 k [ (cos(kx-ωt) + 𝒊 sin(kx-ωt)]
𝝏Ψ(x,t)
𝝏𝒙
= 𝒊 k Ψ(x,t)
𝝏Ψ(x,t)
𝝏𝒙
= 𝒊
𝒑
ħ
Ψ(x,t) { k =
𝟐𝝅
𝝀
=
𝟐𝝅
𝒉
P }
ħ
𝒊
𝝏Ψ(x,t)
𝝏𝒙
= p Ψ(x,t) { k =
𝑷
ħ
}
- 𝒊 ħ
𝝏Ψ(x,t)
𝝏𝒙
= p Ψ(x,t)
- 𝒊 ħ ∂∕∂x [Ψ(x,t)]= p [Ψ(x,t)]

10. Normalization of wave
function
 Let P(x) is a probability function of a particle in a state Ψ(r).
Probability of finding the particle in a small volume dτ = dxdydz
P(r)dτ = dxdydz
∫v
P(x)dτ = probability of finding the particle in volume v.
∫v
P(x)dτ = 1 {When the integration is taken over whole space}
∫v
P(x)dx = 1 {When the quantum particle is bounded in a certain region and have
no chance of escaping}.
According to Max Born’s statistical interpretation of wave function,The probability P(r)
of finding the particle r at a given time t is proportional to |Ψ|2
or ΨΨ*.

11. P(r) =
𝜳𝜳∗
𝜳𝜳∗𝒅𝑻
Here if we multiply or divide Ψ by any constant P(r) will remain same.To find
that constant that makes denominator 1 is simply the normalization of wave function.

12. Examples: