Wave function

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 Comparing (1) and (2),we get, m + kx = 0 After solving we get a result, x(t) = A(cos ωt - Φ ) Mobile Password, ID Card  

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)> = <P> =  

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- ω t) + i sin(kx- ω t)

9. = -k sin ( kx-ωt ) + kcos ( kx-ωt ) = k [ (cos( kx-ωt ) + sin( kx-ωt )] = k Ψ( x,t ) = Ψ( x,t ) { k = = P } = p Ψ( x,t ) { k = } - ħ = p Ψ( x,t ) - 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 d τ = dxdydz P(r)d τ = dxdydz ∫ v P(x)d τ = probability of finding the particle in volume v. ∫ v P(x)d τ = 1 {When the integration is taken over whole space} ∫ v P(x)d x = 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) = 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: