Modern Physicslecture X

Louis de Broglie1892 - 1987

Wave Properties of Matter In 1923 Louis de Broglie postulated that perhaps matter In 1923 Louis de Broglie postulated that perhaps matter

exhibits the same “duality” that light exhibitsexhibits the same “duality” that light exhibits Perhaps all matter has both characteristics as wellPerhaps all matter has both characteristics as well Previously we saw that, for photons,Previously we saw that, for photons,

h

c

hf

c

Ep

mv

h

p

h

Which says that the wavelength of light is related to its Which says that the wavelength of light is related to its momentummomentum

Making the same comparison for matter we find…Making the same comparison for matter we find…

Quantum mechanics

Wave-particle dualityWave-particle duality Waves and particles have interchangeable propertiesWaves and particles have interchangeable properties This is an example of a system with This is an example of a system with complementary complementary

propertiesproperties

The mechanics for dealing with systems The mechanics for dealing with systems when these properties become important is when these properties become important is called “Quantum Mechanics”called “Quantum Mechanics”

The Uncertainty Principle

Measurement disturbes the system

The Uncertainty Principle Classical physicsClassical physics

Measurement uncertainty is due to limitations of the measurement Measurement uncertainty is due to limitations of the measurement apparatusapparatus

There is no limit in principle to how accurate a measurement can There is no limit in principle to how accurate a measurement can be madebe made

Quantum MechanicsQuantum Mechanics There is a fundamental limit to the accuracy of a measurement There is a fundamental limit to the accuracy of a measurement

determined by the determined by the HeisenbHeisenbeerg uncertainty principlerg uncertainty principle If a measurement of position is made with precision If a measurement of position is made with precision x and a x and a

simultaneous measurement of linear momentum is made with simultaneous measurement of linear momentum is made with precision precision ppxx, then the product of the two uncertainties can never , then the product of the two uncertainties can never be less than h/4be less than h/4

2/ xpx

Energy and time

2/ E

Uncertainty principle

The Uncertainty Principle In other words:In other words:

It is physically impossible to measure simultaneously the exact It is physically impossible to measure simultaneously the exact position and linear momentum of a particle position and linear momentum of a particle

These properties are called “complementary”These properties are called “complementary” That is only the value of one property can be known at a timeThat is only the value of one property can be known at a time Some examples of complementary properties are Some examples of complementary properties are

Which way / Interference in a double slit experimentWhich way / Interference in a double slit experiment Position / Momentum (Position / Momentum (xxp > h/4p > h/4)) Energy / Time (Energy / Time (EEt > h/4t > h/4)) Amplitude / PhaseAmplitude / Phase

Erwin Schrödinger1887 - 1961

Wave equations for probabilities In 1926 Erwin Schroedinger proposed a wave In 1926 Erwin Schroedinger proposed a wave

equation that describes how matter waves (or the equation that describes how matter waves (or the wave function) propagate in space and timewave function) propagate in space and time

The wave function contains all of the information The wave function contains all of the information that can be known about a particlethat can be known about a particle

)(

222

2

UEm

dx

d

Wave Function In quantum mechanics, matter waves are In quantum mechanics, matter waves are

described by a complex valued described by a complex valued wave functionwave function, , The absolute square gives the probability of The absolute square gives the probability of

finding the particle at some point in spacefinding the particle at some point in space

This leads to an interpretation of the double slit This leads to an interpretation of the double slit experimentexperiment

*2

Wave functions The wave function of a free particle moving The wave function of a free particle moving

along the x-axis is given byalong the x-axis is given by

This represents a snap-shot of the wave This represents a snap-shot of the wave function at a particular time function at a particular time

We cannot, however, measure We cannot, however, measure , we can , we can only measure |only measure |||22, the , the probability densityprobability density

kxAx

Ax sin2

sin

Max Born

Interpretation of the Wavefunction Max Born suggested that Max Born suggested that was the was the PROBABILITY PROBABILITY

AMPLITUDE AMPLITUDE of finding the particle per unit volumeof finding the particle per unit volume ThusThus

||||22dVdV = =dVdV ((designates complex conjugatedesignates complex conjugate)) is the probability of is the probability of

finding the particle within the volume finding the particle within the volume dVdV The quantityThe quantity | |||22is calledis called the PROBABILITY the PROBABILITY

DENSITYDENSITY Since the chance of finding the particle somewhere in Since the chance of finding the particle somewhere in

space is unity we havespace is unity we have

12

dVψdVψ*ψ

• When this condition is satisfied we say that the wavefunction is NORMALISED

0

2sin)()(

d

xAx

A particle or a wave?

small px small and x big

big px big and x small

Schrödinger Wave Equation The Schrödinger wave equation is one of the most The Schrödinger wave equation is one of the most

powerful techniques for solving problems in powerful techniques for solving problems in quantum physicsquantum physics

In general the equation is applied in three In general the equation is applied in three dimensions of space as well as timedimensions of space as well as time

For simplicity we will consider only the one For simplicity we will consider only the one dimensional, time independent casedimensional, time independent case

The wave equation for a wave of displacement The wave equation for a wave of displacement yy and velocity and velocity vv is given by is given by

2

2

22

2 1

t

y

vx

y

Solution to the Wave equation

We consider a trial solution by substitutingWe consider a trial solution by substituting

yy ((xx,, tt ) = ) = ((xx ) sin() sin( tt ))

into the wave equationinto the wave equation

2

2

22

2 1

t

y

vx

y

• By making this substitution we find that

ψv

ω

x

ψ2

2

2

2

• Where /v = 2/ and p = h/• Thus

/v 2(2/)

Energy and the Schrödinger Equation Consider the total energyConsider the total energy

Total energy Total energy EE = Kinetic energy + Potential Energy = Kinetic energy + Potential Energy EE = = mm vv

22/2/2 ++UU EE = = pp

/(2/(2mm )) ++UU Reorganise equation to giveReorganise equation to give

pp 22

== 22 mm ((EE - - UU )) From equation on previous slide we get From equation on previous slide we get

UEm

v

ω

22

2 2

• Going back to the wave equation we have

02

22

2

ψUEm

x

ψ

• This is the time-independent Schrödinger wave equation in one dimension

Solution to the SWE The solutions The solutions ((xx) are called the STATIONARY STATES of the ) are called the STATIONARY STATES of the

systemsystem The equation is solved by imposing BOUNDARY CONDITIONSThe equation is solved by imposing BOUNDARY CONDITIONS The imposition of these conditions leads naturally to energy levelsThe imposition of these conditions leads naturally to energy levels If we set If we set

r

e

πεU

2

04

1

We get the same results as Bohr for the energy levels of the one electron atomThe SWE gives a very general way of solving problems in quantum physics

Probability and Quantum Physics In quantum physics (or quantum mechanics) we deal with In quantum physics (or quantum mechanics) we deal with

probabilities of particles being at some point in space at probabilities of particles being at some point in space at some timesome time

We cannot specify the precise location of the particle in We cannot specify the precise location of the particle in space and timespace and time

We deal with averages of physical propertiesWe deal with averages of physical properties Particles passing through a slit will form a diffraction patternParticles passing through a slit will form a diffraction pattern Any given particle can fall at any point on the receiving Any given particle can fall at any point on the receiving

screenscreen It is only by building up a picture based on many It is only by building up a picture based on many

observations that we can produce a clear diffraction patternobservations that we can produce a clear diffraction pattern

Wave Mechanics We can solve very simple problems in quantum physics We can solve very simple problems in quantum physics

using the SWEusing the SWE This is sometimes called WAVE MECHANICSThis is sometimes called WAVE MECHANICS There are very few problems that can be solved exactlyThere are very few problems that can be solved exactly Approximation methods have to be usedApproximation methods have to be used The simplest problem that we can solve is that of a particle The simplest problem that we can solve is that of a particle

in a boxin a box This is sometimes called a particle in an infinite potential This is sometimes called a particle in an infinite potential

wellwell This problem has recently become significant as it can be This problem has recently become significant as it can be

applied to laser diodes like the ones used in CD playersapplied to laser diodes like the ones used in CD players