matter waves wave function quantum mechanics

7/28/2019 Matter Waves Wave Function Quantum Mechanics

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Quantum MechanicsLecture-7

Reference: Concept of Modern Physics

by A. Beiser


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Matter Waves...Quantitative ; Wave Function

The wave associated with particles in motion aremathematical constructs. It does not describe thespace time variation of any measurable quantitylike displacement or any other characteristics

present in the medium.

The wave relates to the probabilities [ |(x,y,z,t)|2 ]

of observing the particle at different space locationsas a function of time.


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Physical interpretation of Wave function

The probability P(r,t)dV to find a particle associated with thewavefunction (r,t) within a small volume dV around a point in space

with coordinate rat some instant tis

P(r,t) is the probability density

For one-dimensional case

Wave function may be a complex quantity, but its mod square willalways be a positive quantity

dVtdVtP2

),(),( rr

2P(x, t)dx (x, t) dx

Here |(r,t)|2 = *(r,t)(r,t)

Thus if,= A+iB

then , * = A iBHence *=A2+B2. thus * is always a positive real quantity.


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The probability of finding

a particle somewhere in avolume Vof space is

Since the probability to findparticle anywhere in space is1,we have condition of

normalization

For 1-Dimensional case,the probability of finding

the particle in thearbitrary interval axb is

VV

V dVtdVtPP2

),(),( rr

1),(

2

spaceall

dVtr

dxtxP

dxtxP

everywhere

b

a

ab

2

2

),(

),(

NORMALIZEDWAVE FUNCTION


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Normalization and Normalized Wave Function

dvdvP

vv

2*

NPNdv

v

*

11 * dvN

v

1*

dvN

v

1*

dvNN

v

1*

11 dvv

The process of making probability density unity is called as

normalization and the wave function is called as normalized

wave function. The probability density of a particle in a givenvolume is

The maximum value of P is 1 and minimum value is 0.

Suppose probability of finding a particle in a given volume be N.i. e.

N

1

N

Here is called as a normalizing factor and is called as normalized wave function.


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Quantum Mechanics

The methods of Quantum Mechanics consist infinding the wavefunction associated with aparticle or a system

Once we know this wavefunction we knoweverythingabout the system!


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Physical Interpretation of Wave function : A satisfactoryinterpretation of the wave function associated with a moving particle was given

by Born in 1926. He postulated that the square of the magnitude of thewave function, ||2 (or *, if is complex), evaluated at a particularpoint is proportional to the probability of finding the particle at thatpoint.

A large value of||2 means a large possibility of the particles presence,while a small value of ||2 means the slight possibility of its presence. ||2 iscalled the probability density , and is the probability amplitude.

According to this interpretation, the probability of finding the particle within anelement of volume d is ||2 d. Since the particle is certainly somewhere, theintegral of||2 d over the whole space must be unity, that is

A wave function that obeys this equation is said to be normalized. Every

acceptable wave function must be normalizable.

2

d 1



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Properties of wave function

Besides being normalizable,

An acceptable wave function must fulfill the following requirements:

(i) must be finite everywhere

(ii) must be single valued.(iii) must be continuous and have a

continuous first derivative everywhere.


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Schrdinger's Time-dependent Wave Equation: In 1926, Schrdingerpresented his famous wave equation as development of de-Broglie ideas of the

wave properties of matter. The Schrdinger's equation is the fundamental equationof quantum mechanics in the same sense as the Newtons second law of motion ofclassical mechanics. It is the differential equation for the de Broglie wavesassociated with particles and describes the motion of particles.

The wave function is for a a particle moving freely in the x-direction is given as

=Ae-i(t-x/) (1)

In terms of energy E and momentum p:

Since =E/h (Plancks relation) and =h/p (de Broglie relation)

(x , t) = A e-(2i/h)(Et-px)

Schrdinger's equation of Motion


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(2 i/h )(Et px )

22 2 2(2 i/h )(Et px )

2 2

(x, t) Ae (i)

Differentiating above equation twice w.r.t. x

2 i 4 pAe p (ii)

x h h

Again differentiating eq (i

(2 i/h )(Et px )

2 22

2 2

) once w.r.t. t ,2 i 2 iE

Ae E (iii)t h h

eq (ii) and (iii) give

1 hp = - (iv)

4 x

and

1 ihE= (v)

2 t



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Schrdinger developed the wave equation which can be solvedto find the wavefunction by translating the equation for energyof classical physics into the language of waves

For fixed energy, we obtain the time-independent Schrdingerequation, which describes stationary states

the energy of such states does not change with time n(x) is an eigenfunction or eigenstate Vis a potential function representing the particle interaction with

the environment

22

2

xV x E x2m x

222

x xV x i

2m x t

2pV E

2m



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2 22 iEt/h 2 iEt/h 2 iEt/h

2 2

h d ih 2 iE- e V e e8 m dx 2 h

Schrdinger's time independent wave equation

In many cases, the potential energy of a particle does not depend upon time and is afunction of the position only. When this is the case, the Schrdinger equation maybesimplified. Let us start from first equation (in terms of energy and momentum)

(x , t) = A e-(2i/h)(Et-px)

In which variables x and t have been separated. If the spatial part is ( x )=Ae2ipx/h

Then the above equation can be written as ( x , t) = (x) e-2iEt/h

Now, by differentiation twice w.r.t. x ,we get

And differentiation w.r.t. time gives

2 22 iEt /h

2 2

de

x dx

2 iEt /h 2 iEe

t h

2 2

2 2

h ih- V8 m x 2 t


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For the same particle moving in three dimensional space, the equationbecomes

This is Schrdinger's time independent (steady state) wave equation for a

particle in three-dimensional space.

For a free particle V=0. therefore, the Schrdinger's equation for a freeparticle is

This is the steady state or time-independent form of Schrdinger

equation.

Note: Wave function is timeindependent.--------(x).

2 2

2 2

h d- V E8 m dx

2 2

2 2

d 8 m(E V) 0

dx h

Schrdinger's time independent wave equation

Three dimensional space

2 2 2 2

2 2 2 2

8 m(E V) 0

x y z h

22

2

8 m(E V) 0

h

22

2

8 mE 0

h

matter waves wave function quantum mechanics

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