Ch40 Young Freedmanx

8/12/2019 Ch40 Young Freedmanx

1/26

Chapter 40/41: Quantum MechanicsWave Functions & 1DSchrodinger Eq Particle in a Box

Wave functionEnergy levels

Potential Wells/Barriers &Tunneling

The Harmonic Oscillator The H-atom


2/26

Quantum Wave Function

In QM, thematter wave

postulated by de Broglie is described by acomplex-

valued wavefunction ( x,t ) which is the fundamental descriptor for a quantum particle.

x,t

Re/Im ( x,t )

( x,t ) is a complex-valued function of

space and time.

1. Its absolute value squared

gives the probability in finding the particle in an infinitesimal volume d xat time t .

2. For any Q problem:The goal is to find for the particle for all time.Physical interactions involvesoperations ( A) on this wavefunction:

Experimental measurements willinvolve the products,

2( , ) x t dx

( , ) x t

( , ) A x t

( , ) A x t ( , ) x t


3/26

Since is a probability, it has to be normalized !

Wave Function and Probability2 *

( , ) ( , ) ( , ) x t x t x t

is the probability distribution function (or probability density )

In other words, 2

xdx

is the probability in finding the particle inthe interval at time t .[ , ] x x dx

(shaded area)

2( ) ( , ) p x dx x t dx

2( ) ( , ) 1 p x dx x t dx

(At any instance of time t , the particle must be somewhere in space !)

2( , ) x t dx

(Similar to the intensity of an electric field: being proportional to the # of photons.)

2 I E


4/26

The Schrodinger EquationIn Classical Mechanics, we have the Newtons equation which describes thetrajectory x(t ) of a particle:

mF x

In EM, we have the wave equation for the propagation of the E, B fields:

2 2

2 2 2, 1 , E B E B

x c t

(derived from Maxwells eqns)

In QM, Schrodinger equation prescribes the evolution of the wavefunctionfor a particle in time t and space x under the influence of a potential energy U ( x),

2 2

2( , ) ( , )( ) ( , )

2 x t x t

U x x t im x t

(general 1D Schrdinger equation)

U ( x)


5/26

Now, from the de Broglie relations, the energy and momentum of this

quantum free particle must be related to its wave number and angularfrequency through:

Since the reference point for is arbitrary, we can simply take .

Then, the energy E of a free particle will simply be its kinetic energy,

Wave Equation for a Quantum FreeParticle

A free particle has no force acting on it. Equivalently, the potential energy

( )U x

k

0 xF dU x dx ( )U x

( ) 0U x

2 2 221

2 2 2m v p

E mvm m

must be a constant for all x, i.e.,

22h

E hf f

22h h p k

2 2

2

k

m

2

2 p

E m

(non-relativistic)


6/26

We now assume the fundamental sinusoidal form for a wave function of aquantum free particle with mass m, momentum and energy :


Thus, a correct quantum wave function for this free particle must satisfy thisquantum dispersion relation for and : k

E

2 2

*

2

k

m

p k

(non-relativistic)

, cos sini kx t

x t Ae A kx t i kx t


7/26


Substituting the trial solution into the Schrodinger'sEquation, we have,

22 2

22

,2 2

i kx t x t A ik em x m

, i kx t x t i i A i e

t

, i kx t x t Ae

2 2

2i kx t k Ae

m

i kx t Ae

LHS:

RHS:

( for a free particle)

( ) 0U x


8/26


Thus, yes indeed! The proposed quantum wave function for a free particle

222

, ,2

x t x t i

m x t

is a solution of the 1D Schrodingers Equation,

2 2

2k m

, i kx t x t Ae

and it satisfies the required quantum dispersion relation:


9/26

The 1D Schrodinger Equation

As we have see,

- the first term (2nd order spatial derivative term) in the Schrodinger equationis associated with the Kinetic Energy of the particle

- the last term (the 1st order time derivative term) is associated with the totalenergy of the particle

- together with the Potential Energy term U ( x) ( x)

the Schrodinger equation is basically a statement on the conservation ofenergy.

KE PE Total E+ =

2 2

2( , ) ( , )( ) ( , )

2 x t x t

U x x t im x t

In fact, in general, we have,


10/26

Stationary StatesFor most problems, we can factor out the time dependence by assuming thefollowing harmonic form for the time dependence,

( , ) ( ) i t x t x e

(Recall the free particle case: .)( , ) ikx i t x t Ae e

With , we can rewrite the time exponent in terms of E ,/( , ) ( ) i E t x t x e

/ E

is a state with a definite energy E and is called a stationary state .( , ) x t

Note that, 2 * * / /( , ) ( , ) ( , ) ( ) ( )iEt iEt x t x t x t x e x e

2* ( / / ) *( ) ( ) ( ) ( ) ( )i Et Et x x e x x x

( ) x is called the time-independent wave function.


11/26

The Time-Independent SchrdingerEquationSubstituting this factorization into the general time-dependent SchrodingerEq, we have

/ / /( , ) ( () )( )iEt iE iE t t x t iE i i i x x e E xe et t

and,2 2

2 2/( , ) ( ) iEt x t d x e

x dx

RHS

LHS

2/

2

2( )

2

iEt d x e

dxm

/( )( ) iEt x eU x /( ) iEt E x e

2 2

2( ) ( ) ( ) ( )

2d x

U x x E xm dx

(time dependence can

be cancelled out !)


12/26

More on Wavefunction

In general, the probability in finding the particle in the interval [a,b ] is given by:2

( )b

ab a p x dx

Note: is not the probability densityis the probability density.

( ) x

Other physical observables can be obtained from ( x) by the following operation:

2( ) x a b x

p(x)

example (position x):2( ) ( ) x xp x dx x x dx

is called the expectation value (of x): it is the experimentalvalue that one should expect to measure in real experiments !

x


13/26

More on Wavefunction

In general, any experimental observable (position, momentum, energy, etc.)O( x) will have an expectation value given by:

2( ) ( )O O x x dx

Expectation values of physically measurable functions arethe only experimentally accessible quantities in QM.

Wavefunction itself is not a physically measureablequantity.

( ) x

Note:

O can be x, p, E , etc.


14/26

Standard Procedure in Solving QMProblems with Schrodinger Equation(with the time-independent Schrodinger Equation)

Given: A particle is moving under the influence of a potential U ( x).

Examples: Free particle: U ( x) = 0

Particle in a box:

Barrier:

HMO:

0, 0( )

, x L

U xelsewhere

0 , 0( )0,

U x LU x

elsewhere

21( ) '2U x k x


15/26

Standard Procedure in Solving QMProblems with Schrodinger EquationSolve time-independent Schrodinger equation for ( x) as a function of energy E ,

with the restrictions:

and are continuous everywhere for smooth U ( x).

is normalized , i.e.,

Bounded solution:

( ) x ( )d xdx

( ) x 2

( ) 1 x dx

( ) 0 x as x

Then, expectation values of physical measurable quantities can be calculated.


16/26

Particle in a Box

A 1- D box with hard walls:

(0) ( )U U L (non-penetrable)

A free particle inside the box:( ) 0U x (inside box)

No forces acting on the particle

except at hard walls.P (in x) is conserved between

bounces|P | is fixed but P switchessign between bounces.

Classical Picture


17/26

Particle in a Box (Quantum Picture)The situation can be described by the following potential energy U ( x):

0, 0( )

, x L

U xelsewhere

2 2

2( ) ( ) ( ) ( )

2d x

U x x E xm dx

The time-independent Schrodinger equation is:

Recall, this is basically

KE PE Total E+ =

Problem statement: For a given U ( x), what are the possible wave functions ( x)and their corresponding energies E ?


18/26

Inside the box, , U ( x) = 0, and the particle is free. From before,we know that the wave function for a free particle has the following form:

Wave functions for a Particle in a Box0 x L

1 2( ) ikx ikx

inside x A e A e (linear combination of the two

possible solutions.)

where A1 and A2 are constants that will be determined later.

Outside the box, , and the particle cannot exist outside the box and ( )U x

( ) 0outside x (outside the box)

At the boundary, x = 0 and x = L,the wavefunction has to be continuous :

(0, ) (0, ) 0inside outside L L

2 2

2

k E

m


19/26

Wavefunctions for a Particle in a BoxLet see how this boundary condition imposes restrictions on the twoconstants, A1 and A2, for the wave function.

Using the Eulers formula, we can rewrite the interior wave function in termsof sine and cosine:

1 2

1 2 1 2

( ) cos sin cos sin

cos sininside x A kx i kx A kx i kx

A A kx i A A kx

Imposing the boundary condition at x = 0,

1 2 1 2(0) cos 0 sin 0inside A A i A A 1 2 0 A A

1 2 A A

1( ) 2 sin sininside x iA kx C kx

(where C =2iA1)

ikxe ikxe


20/26

Wavefunctions for a Particle in a Box Now, consider the boundary condition at x = L:

( ) sin 0inside L C kL

For a non-trivial solution ( ), only certain sine waves with a particular choice of wave numbers (k ) can satisfy this condition:

0C

or , 1, 2,3,n nn

k L n k n L

This implies that the wavelengths within the box is quantized !

2 2nnk

Ln

2 , 1, 2,3, L

nn

Allowed wavelengths are the ones which can fit within the box !


21/26

Wavefunction for a Particle in a Box

2 2 , 1,2,3,nn

L nk n

/ 2

3 / 2

L must fit an integral number ofhalf-wavelengths: 2n L n

Then, ( ) sin sin ,

1,2,3,

n n

nk

L x C x C x

n

(similar to standing waves on a cramped string)

2

5 / 2

, 1, 2,3,nn

k n L

Or,


22/26

Quantized Energies for a Particle in aBoxSince the wave number k n is quantized, the energy for the particle in the box isalso quantized:

22 2 2 2 2 2 2 2

2 2or , 1,2,3,2 2 2 8n

n

k n n n h E n

m m L mL mL

Note: the lowest energyis not zero:

2

1 2 08h

E mL

n = 0 gives ( x) = 0and it means no particle.

(n is called thequantum number )


23/26

Probability and WavefunctionRecall that | ( x)|2 (and not the wavefunction itself ( x)) is the probabilitydensity function. In particular,

2 2 2( ) sin n x

x dx C dx L

gives the probability in finding the particle in an interval [ x, x+dx] within the box.


24/26

Probability in Finding the Particle Notes:

The positions for the particles are probabilistic . We just know that ithas to be in the box but the exactlocation within the box is uncertain.

Not all positions between x = 0 and L are equally likely. In CM, all positions are equally likely for the particle in the box.

There are positions where the particle has zero probability to befound.


25/26

Probability and Normalization

Although we dont know exactly where the particle might be inside the box,we know that it has to be in the box. This means that,

2

( ) 1 x dx

2 2 2

0sin 1

2 L n L

C x dx C L

(normalization condition)

So, the normalization condition fixes the final free constant C in the

wavefunction, . This then gives,2C L

2( ) sinnn x

x L L

(particle in a box)


26/26

Time Dependence Note that with ( x) found, we can write down the full wavefunction for thetime-dependent Schrodinger equation as:

/( , ) ( ) iEt n n x t x e

Note that the absolute value for is unity , i.e.,

/2( , ) sin iEt n

n x x t e

L L

/iEt e 2/ / / 0 1iEt iEt iEt e e e e

so that | n( x,t )|2 = | ( x)|2 is independent of time and probability density in

finding the particle in the box is also independent of time.

recall E hf

Documents

Ch40 Young Freedmanx