Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that,...

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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”).

Transcript of Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that,...

Page 1: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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”).

Page 2: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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.

Page 3: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

Page 4: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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)

),,,,,,( 222111 tzyxzyx

d *

Page 5: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

Page 6: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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.

Page 7: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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.

Page 8: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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.

Page 9: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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.

Page 10: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Acceptable or Not ??

Page 11: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Acceptable or not acceptable ??

xiv

x

xiii

eii

ei

x

x

1sin)(

sin)(

),()(

),0()(

Page 12: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Exp(x)

Page 13: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Sinx/x

Page 14: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Sin-1x

Page 15: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Postulate 2

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

Page 16: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Operator

• A rule that transforms a given function into another function

Page 17: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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).

Page 18: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Identifying the operators

Page 19: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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.

Page 20: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Hermitian operator

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

satisfies A operator hermitianA ˆ

Page 21: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

• Prove Operator x is Hermitian.

Page 22: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

Page 23: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Hermitian operator or not ??

2

2

)(

)(

)(

xiii

xiii

xi

Page 24: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Linear Operator

• A linear operator has the following properties

fccf

ffff

AA

AAA 2121

Page 25: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Linear operator

Derivative

integrals

log

Page 26: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Normalized wave function:

Orthogonal wave functions:

Orthonormal set wave functions:* 1, if

0, if

m n mndx m n

m n

1*2 dxN

Page 27: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Hermitian operator

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

Page 28: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

dxdx

d

idx

dx

d

ip rssrsrrsx

***)(

Page 29: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

Page 30: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Eigen Function and Eigen value

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

)()(A

xkfxf

Page 31: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

Page 32: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Eigen function and eigen value

ikxexf Is it eigen function of momentum operator ?

What is eigen value ?

Page 33: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

Page 34: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

Page 35: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Commute or not ??

• Operator x and d/dx

Page 36: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

They don’t

1BA,

)()(BA,

)()()]([)(AB

)()(BA

B

xA

,

,

xfxf

xfxxfxxfdx

dxf

xxfxf

dx

d

Page 37: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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:

Page 38: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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.

Page 39: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Postulate 5

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

Page 40: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

Schrodinger Equation

Time independent Schrodinger equation

General form:

H = E

E= T + V

Page 41: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

p ix

recall

Copyright – Michael D. Fayer, 2007

Page 42: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

Page 43: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

( )( , , ) ( , , )

( ) ( , , )

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

Page 44: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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

Page 45: Wavefunction Quantum mechanics acknowledges the wave- particle duality of matter by supposing that, rather than traveling along a definite path, a particle.

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