Lesson 4 SWE

8/13/2019 Lesson 4 SWE

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Quantum Mechanics for

Scientists and Engineers

David Miller


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Schrdingers equation

From de Broglie to Schrdinger


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Electrons as waves

de Broglies hypothesis is that the electron

wavelength is given by

wherep is the electron momentum andh is Plancks constant

J s

Now we want to use this to help construct a

wave equation

h

p

346.62606957 10h


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A Helmholtz wave equation

If we are considering only waves of one

wavelength for the momenti.e., monochromatic waves

we can choose a Helmholtz wave equation

with

which we know works for simple waves

with solutions like

sin(kz), cos(kz), and exp(ikz)(and sin(kz), cos(kz), and exp(ikz))

2

22d k

dz 2k


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A Helmholtz wave equation

In three dimensions, we can write this as

which has solutions likesin(k r), cos(k r), and exp(ik r)(and sin(-k r), cos(-k r), and exp(-ik r))

where k and r are vectors

2 2 22 2

2 2 2 k

x y z


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From Helmholtz to Schrdinger

With de Broglies hypothesis

and the definitionthen

where we have defined

soHence we can rewrite our Helmholtz equation

or

/h p

2 /k

2 2 2

/k p

2 / /k p h p

/ 2h

22

2

p

2 2 2p


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If we are thinking of an electron, we can

divide both sides by its mass mo to obtain

But we know from classical mechanics that

and in general

2 22

2 2o o

p

m m

2

2kinetic energy of electron

o

p

m

Total energy ( )=Kinetic energy + Potential energy ( )E V r


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So

Hence our Helmholtz equation

becomes the Schrdinger equation

or equivalently

2 22

2 2o o

p

m m

2 / 2Kinetic energy =

= Total energy ( ) - Potential energy ( )

op m

E V r

2

2

2o

E Vm

r

2

2

2o

V Em

r


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Schrdingers time-independent equation

We can postulate a Schrdinger equation for

any particle of mass m

Formally, this is the

time-independent Schrdinger equation

2

2

2

V E

m

r


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Probability densities

Borns postulate is that

the probability of finding an electronnear any specific point r in space

is proportional to the modulus squared

of the wave amplitudecan therefore be viewed as a

probability density

with called a probability amplitude

or a quantum mechanical amplitude

P r

2

r

r

2

r

r


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Diffraction by two slits


12/25

Youngs slits

An opaque mask has two slits cut in it, a distance s apart

s


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Youngs slits

We shine a plane wave on the mask from the left

s


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Youngs slits

What will be the pattern on a screen at a large distancezo?

s

oz

?


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Youngs slits

The slits as point sources give an interference pattern


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Youngs slits

The distance from the upper source to point x

on the screen is 2

2/ 2 ox s z

oz

2

s

x/ 2x s

2 22 2/ 2 1 / 2 /

o o ox s z z x s z

2/ 2 / 2o oz x s z 2 2/ 2 / 8 / 2

o o o oz x z s z sx z


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Youngs slits

The distance from the lower source to point x

on the screen is 2 2

/ 2 ox s z

oz

2

s

x

/ 2x s

2

/ 2 / 2o o

z x s z 2 2/ 2 / 8 / 2

o o o oz x z s z sx z


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Youngs slits

For large zo the waves are approximately uniformly bright

i.e., using exponential waves for convenience

Using our approximate formulas for the distances gives

where

2 22 2exp / 2 exp / 2

s o ox ik x s z ik x s z

exp exp / 2 exp / 2s o ox i ik sx z ik sx z 2 2/ 2 / 8o o ok z x z s z


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Youngs slits

Now

so

so the intensity of the beam

exp exp 2cosi i

exp exp exp2 2

s

o o

sx sxx i ik ik

z z

2

2 1

cos / 1 cos 2 /2

s o ox sx z sx z

exp cos exp cos2

o o

sx sxi k iz z


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Youngs slits

The interference fringes are spaced by /s od z s

sd

s

oz


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Youngs slits

This allows us to measure small wavelengths /s od s z

sd

s

oz


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Interpreting diffraction by two slits


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Youngs slits

If the upper slit is blocked no interference pattern


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Youngs slits

If the lower slit is blocked no interference pattern


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Lesson 4 SWE