schrodinger equation

7/27/2019 Schrodinger Equation

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Schrodinger Equation


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Each particleis represented by awave function (x,t) such that *=theprobability of finding the particle at that

position at that time

The wave function is used in theSchrodinger equation. The Schrodinger

equation plays the role of Newtons lawsand conservation of energy in classicalmechanics. It predicts analytically andprecisely the probability of events.

THE WAVE FUNCTION


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Wavefunction Properties

contains all the measurable information aboutthe particle

* summed over all space=1(if particleexists, probability of finding it somewheremust be one and its derivative is continuous

allows energy calculations via the Schrodingerequation

permits calculation of expectation value of agiven variable.

is finite and single valued


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))(/( pxEtiAe

SCHRODINGERS EQUATION:

TIME-DEPENDENT FORM

The wave function for a particle moving freely in

the +x direction is given by.(1)

where E is total energy of the particle and p is its

momentum.Differentiating (1) for twice w r to x gives

Also differentiating (1) for once w r to tgives

2

2

22

2

p

xor

.(2)

iE

t .(3)or tiE


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),(2

2

txVm

pE

V

xmt

i2

22

2

.(4)

The quantities

are called operators.

The total energy E of aparticle is the sum of itskinetic energy and potentialenergy

Multiply (5) by on bothsides gives

Substitute for E and p2 to obtain the timedependent form of schrodinger equation:

.(6)

..(5)

.(7)

tiE

and

Vm

pE2

2


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V

zyxmti

2

2

2

2

2

22

2

In three dimensions the time dependent form ofSchrodingers equation is

.(8)

Where the particles potential energy V is some

function of x,y,z and t. Any restriction that maybe present on the particles motion will affect thepotential energy and once V is knownSchrodinger's equation may be solved to obtain the

energy, wave function and probability density ofthe particle.


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EXPECTATION VALUES

)1...(..................

.....

321

332211

i

ii

N

xN

NNN

xNxNxNX

)2.....(....................2dxP ii

The average position of a number of particlesdistributed along the x axis such that N1 particles

at x1, N2 particles at x2 and so on is given by

For a single particle the number Ni at xi must bereplaced by the probability

Where i is the particle wave function evaluatedat x=xi. Making substitution and changing thesummations to integrals, the expectation value ofthe position is given by


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)3..(..............................2

2

dx

dxx

x

If the wave function is normalized

then )4...(..............................2

dxxx

1..2

dxei

)5......(2

dxEE

dxpp2

Similarly expectation values of momentum andenergy are given by

and


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)1(.......................................... )/(

)/()/())(/(

tiE

xiptiEpxEti

eei

eAeAe

)2(..........2 2

22

V

xmti

tiEtiEtiE

eVxemeE

)/(

2

2)/(

2)/(

2

xipAewhere )/(

SCHRODINGER EQUATION: STEADY-STATE FORM

The wave function

of unrestricted particle may bewritten as

is the product of time dependent and a positiondependent function

Substituting in time dependent form ofSchrodinger equation

We get


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Dividing through by the common exponential factorgives

is the steady-state form of Schrodinger equation.

In three dimensions the equation takes the form as

)4........(0)(2

22

2

2

2

2

2

VE

m

zyx

)3(....................0)(

222

2

VE

m

x


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EIGENVALUES AND EIGENFUNCTIONS

The values of energy En for which Schrodingers

steady-state equation can be solved are calledeigenvalues and corresponding wave functions nare called eigenfunctions.

The discrete energy levels of the hydrogen atom

are an example of a set of eigenvalues

2222

4 1

32 n

meEn

n = 1,2,3,..


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)1.(..............................02

22

2

E

m

x

APPLICATION: PARTICLE IN A BOX

0 L x

V

Aim: To show energy quantization

1.How to solve Schrodingers equation?

2.Obtain eigenvalues and eigenfunctions

3.Compare the results with classicalmechanics

The potential energy of the particle isV=0 inside the box andV= outside the box

The steady-state form of Schrodingersequation for the particle within the box with aboverestrictions may be written as


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)2...(2

sin xmE

A

)4......(..............................2 nLmE

The equation (1) will have the following possiblesolutions

)3......(2

cos xmE

B

and

These solutions are subject to the importantboundary condition such that =0 for x=0 and x=LSince cos(0)=1, the second solution cannot

describe the particle at x=0 implies B=0

The first solution gives=0 for x=0 but will bezero at x=L only when

n=1,2,3,..

From equation (4) the energy of the particle canhave only certain values called eigenvalues givenby


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)7...(..........)sin(2

sin xL

nAx

mEA

n

n

)5(..............................2 2

222

mL

nEn

n=1,2,3,..

A particle confined to a box cannot have an

arbitrary energy and also n=0 or E=0 is notadmissible

WAVE FUNCTIONS

The wave function of a particle in a box whoseenergy is E is

Substituting for E from equation (5) we get

)6.....(..........2

sin xmE

A

further )8.........(....................1

2

dx


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)10..(..........sin2

xL

n

Ln

)9.(..............................2

LA

gives

Therefore the normalized wavefunctions of theparticle are

The normalized wanefunctions

1,2, 3 are plotted as shown.


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Graphical explanation

A particle in a box with infinitely high

walls is an application of the Schrodingerequation which yields some insights intoparticle confinement. The wavefunctionmust be zero at the walls and the solution

for the wavefunction yields just sine waves.

The longest wavelength is = 2L andthe higher

modes have wavelengths given by

=2L/n where n= 1,2,3,.


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When this is substituted into the DeBroglierelationship it yields momentum p=h/

=nh/2LThe momentum expression for the particlein a box :

p=h/ =nh/2L: n= 1,2,3,..is used to calculate the energy associatedwith the particle

nEmL

hn

m

pmv

2

2222

822

1


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Though oversimplified, this indicates someimportant things about bound states for

particles:

1. The energies are quantized and can becharacterized by a quantum number n

2. The energy cannot be exactly zero.

3. The smaller the confinement, the largerthe energy required.

If ti l i fi d i t


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If a particle is confined into arectangular volume, the same kind ofprocess can be applied to a three-

dimensional "particle in a box", and thesame kind of energy contribution is madefrom each dimension. The energies for athree-dimensional box are

This gives a more physically realisticexpression for the available energies forcontained particles. This expression is usedin determining the density of possible

energy states for electrons in solids.

2

2

3

2

2

2

1

8mLE

nnnn
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