Wavefunction

• Quantum mechanics acknowledges the wave-particle duality of matter by supposing that, rather than traveling along a definite path, a particle is distributed through space like a wave. The wave that in quantum mechanics replaces the classical concept of particle trajectory is called a wavefunction, ψ (“psi”).

A wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles. There is one wave function containing all the information about the entire system, not a separate wave function for each particle in the system.

Wave equation for the harmonic motion

)()()(

8

]2[

]2[

22

1

)()(

4

)(2)(

2

2

2

2

2

1

2

1

22

2

2

2

2

2

2

2

xVEdx

xd

m

h

VEm

h

p

h

VEmp

Vm

pVmvE

xdx

xd

xdx

xd

Postulates of Quantum Mechanics

Postulate 1:

State and wave functions. Born interpretation

The state of a quantum mechanical system is completely specified by a wave function ψ (r,t) that depends on the coordinates of the particles (r) and time t. These functions are called wave functions or state functions.For 2 particle system:

Wave function contains all the information about a system. wave function classical trajectory

(Quantum mechanics) (Newtonian mechanics)

Meaning of wave function:

P(r) = |ψ|2 = => the probability that the particle can be found at a particular point x and a particular time t. (Born’s / Copenhagen interpretation)

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d *

Implications of Born’s Interpretation

(1) Positivity:

P(r) >= 0

The sign of a wavefunction has no direct physical significance:

The positive and negative regions of this wavefunction both

correspond to the same probability distribution.

(2) Normalization:

i.e. the probability of finding the particle in the universe is 1.

spaceall

d_

* 1

Physically acceptable wave function

The wave function and its first derivative must be:

1) Finite. The wave function must be single valued. This means that for any given values of x and t , Ψ(x,t) must have a unique value. This is a way of guaranteeing that there is only a single value for the probability of the system being in a given state.

2. Square-integrable

The wave function must be square-integrable. In other words, the integral of |Ψ|2 over all space must be finite. This is another way of saying that it must be possible to use |Ψ|2 as a probability density, since any probability density must integrate over all space to give a value of 1, which is clearly not possible if the integral of |Ψ|2 is infinite. One consequence of this proposal is that must tend to 0 for infinite distances.

Continuous wavefunction

• A rapid change would mean that the derivative of the function was very large (either a very large positive or negative number). In the limit of a step function, this would imply an infinite derivative. Since the momentum of the system is found using the momentum operator, which is a first order derivative, this would imply an infinite momentum, which is not possible in a physically realistic system.

Continuous First derivative

1. All first-order derivatives of the wave function must be continuous. Following the same reasoning as in condition 3, a discontinuous first derivative would imply an infinite second derivative, and since the energy of the system is found using the second derivative, a discontinuous first derivative would imply an infinite energy, which again is not physically realistic.

Acceptable or Not ??

Acceptable or not acceptable ??

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x

xiii

eii

ei

x

x

1sin)(

sin)(

),()(

),0()(

Exp(x)

Sinx/x

Sin-1x

Postulate 2

To every physical property, observable in classical mechanics, there corresponds a linear, hermitian operator in quantum mechanics.

Operator

• A rule that transforms a given function into another function

Operator

• Example. Apply the following operators on the given functions:

• (a) Operator d/dx and function x2.• (b) Operator d2/dx2 and function 4x2.• (c) Operator (∂/∂y)x and function xy2.• (d) Operator −iћd/dx and function exp(−ikx).• (e) Operator −ћ2d2/dx2 and function exp(−ikx).

Identifying the operators

Hermitian Operator

• Hermitian operators have two properties that forms the basis of quantum mechanics

(i) Eigen value of a Hermitian operator are real.(ii) Eigenfunctions of Hermitian operators are

orthogonal to each other or can be made orthogonal by taking linear combinations of them.

Hermitian operator

behaved wellare g and f if ;dx f)Ag(dxgAf **

satisfies A operator hermitianA ˆ

• Prove Operator x is Hermitian.

******** )( srrssrsrrs xdxxdxxdxxx

Hermitian operator or not ??

2

2

)(

)(

)(

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xiii

xi

Linear Operator

• A linear operator has the following properties

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ffff

AA

AAA 2121

Linear operator

Derivative

integrals

log

√

Normalized wave function:

Orthogonal wave functions:

Orthonormal set wave functions:* 1, if

0, if

m n mndx m n

m n

1*2 dxN

Hermitian operator

• Example. Prove that the momentum operator (in one dimension) is Hermitian.

dxdx

d

idx

dx

d

ip rssrsrrsx

***)(

Postulate 3

In any measurement of the observable associated with the operator , the only values that will ever be observed are the eigenvalues ‘a’ which satisfy the eigenvalue equation:

This is the postulate that the values of dynamical variables are quantized in quantum mechanics.

A

Eigen Function and Eigen value

k eeigen valu with A ofion eigenfunct is f(x)

)()(A

xkfxf

Q: What are the eigenfunctions and eigenvalues of the operator d/dx ?

Eigen function and eigen value

ikxexf Is it eigen function of momentum operator ?

What is eigen value ?

Eigenvalue equation

Eigenvalue equation

(Operator)(function) = (constant factor)*(same function)

Example: eikx is an eigenfunction of a operator Px = -ih x

-i2 k2eikx= h

F(x) = eikx

h x

eikx= -i

h k2eikx= Thus eikx is an eigenfunction

Significance of commutation rules

• The eigenvalue of a Hermitian operator is real.• A real eigenvalue means that the physical quantity for which the operator

stands for can be measured experimentally.• The eigenvalues of two commuting operators can be computed by using

the common set of eigenfunctions.

If the two operators commute, then it is possible to measure the simultaneously the precise value of both the physical quantities for which the operators stand for.

Question: Find commutator of the operators x and px

Is it expected to be a non-zero or zero quantity?

Hint: Heisenberg Uncertainty Principle

Commute or not ??

• Operator x and d/dx

They don’t

1BA,

)()(BA,

)()()]([)(AB

)()(BA

B

xA

,

,

xfxf

xfxxfxxfdx

dxf

xxfxf

dx

d

Postulate 4

For a system in a state described by a normalized wave function , the average or expectation value of the observable corresponding to A is given by:

Mean value theoremExpectation value in general:

The fourth postulates states what will be measured when large number of identical systems are interrogated one time. Only after large number of measurements will it converge to <a>.

In QM, the act of the measurement causes the system to “collapse” into a single eigenstate and in the absence of an external perturbation it will remain in that eigenstate.

Postulate 5

The wave function of a system evolves in time in accordance with the time dependent Schrodinger equation:

Schrodinger Equation

Time independent Schrodinger equation

General form:

H = E

E= T + V

Schrödinger Representation – Schrödinger Equation

2

2classical

pH V

m

Hamiltonian kinetic potential energy energy

Sum of kinetic energy and potential energy.

Time dependent Schrödinger Equation

( , , , )( , , , ) ( , , , )

x y z ti H x y z t x y z t

t

Developed through analogy to Maxwell’s equations and knowledge ofthe Bohr model of the H atom.

The potential, V, makes one problem different form another H atom, harmonic oscillator.

Q.M. 2 2

2(x)

2m xH V

22 ( , , )

2H V x y z

m

2 2 22

2 2 2x y z

one dimension

three dimensions

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recall

Copyright – Michael D. Fayer, 2007

Getting the Time Independent Schrödinger Equation

( , , , )x y z t wavefunction

( , , , ) ( , , , ) ( , , , )i x y z t H x y z t x y z tt

( , , , ) ( , , ) ( )x y z t x y z F t

If the energy is independent of time ( , , )H x y z

Try solution

product of spatial function and time function

Then

( , , ) ( ) ( , , ) ( , , ) ( )i x y z F t H x y z x y z F tt

( , , ) ( ) ( ) ( , , ) ( , , )i x y z F t F t H x y z x y zt

independent of t independent of x, y, z

divide through by

F

Copyright – Michael D. Fayer, 2007

( )( , , ) ( , , )

( ) ( , , )

dF ti H x y z x y zdt

F t x y z

depends onlyon t

depends onlyon x, y, z

Can only be true for any x, y, z, t if both sides equal a constant.

Changing t on the left doesn’t change the value on the right.

Changing x, y, z on right doesn’t change value on left.

Equal constant

dFi Hdt E

F

Copyright – Michael D. Fayer, 2007

dFi Hdt E

F

( , , ) ( , , ) ( , , )H x y z x y z E x y z

Both sides equal a constant, E.

Energy eigenvalue problem – time independent Schrödinger Equation

H is energy operator.

Operate on get back times a number.

’s are energy eigenkets; eigenfunctions; wavefunctions.

E Energy EigenvaluesObservable values of energy

Copyright – Michael D. Fayer, 2007

Time Dependent Equation (H time independent)

( )

( )

dF ti

dt EF t

( )( )

dF ti E F t

dt

( ).

( )

dF t iE dt

F t

lni Et

F C

/( ) i E t i tF t e e

Time dependent part of wavefunction for time independent Hamiltonian.

Time dependent phase factor used in wave packet problem.

Integrate both sides

Copyright – Michael D. Fayer, 2007