Modern Physicslecture 3

Louis de Broglie1892 - 1987

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

exhibits the same “duality” that light exhibits Perhaps all matter has both characteristics as well 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 momentum

Making the same comparison for matter we find…

Quantum mechanics

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

properties

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

The Uncertainty Principle

Measurement disturbes the system

The Uncertainty Principle Classical physics

Measurement uncertainty is due to limitations of the measurement apparatus

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

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

determined by the Heisenberg uncertainty principle If a measurement of position is made with precision Dx and a

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

2/ xpx

The Uncertainty Principle In other words:

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

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

Which way / Interference in a double slit experiment Position / Momentum (DxDp > h/4p) Energy / Time (DEDt > h/4p) Amplitude / Phase

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

powerful techniques for solving problems in quantum physics

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

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

The wave equation for a wave of displacement y and velocity v is given by

2

2

22

2 1

t

y

vx

y

Erwin Schrödinger1887 - 1961

Solution to the Wave equation

We consider a trial solution by substituting

y (x, t ) = y (x ) sin(w t )

into 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 w /v = 2p/l and p = h/l• Thus

w 2/ v 2 = (2p/l)2

Energy and the Schrödinger Equation Consider the total energyTotal energy E = Kinetic energy + Potential Energy

E = m v 2/2 +U

E = p 2/(2m ) +U

Reorganise equation to givep

2 = 2 m (E - U )

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

Wave equations for probabilities In 1926 Erwin Schroedinger proposed a wave

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

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

)(

222

2

UEm

dx

d

Solution to the SWE The solutions y(x) are called the STATIONARY

STATES of the system The equation is solved by imposing BOUNDARY

CONDITIONS The imposition of these conditions leads naturally

to energy levels 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

Wave Function In quantum mechanics, matter waves are

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

finding the particle at some point in space

This leads to an interpretation of the double slit experiment

*2

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

AMPLITUDE of finding the particle per unit volume Thus

|y |2 dV = y y * dV (y * designates complex conjugate) is the probability of

finding the particle within the volume dV The quantity |y |2 is called the PROBABILITY

DENSITY Since the chance of finding the particle somewhere in

space is unity we have

12

dVψdVψ*ψ

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

Max Born

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

deal with probabilities of particles being at some point in space at some time

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

We deal with averages of physical properties Particles passing through a slit will form a

diffraction pattern Any given particle can fall at any point on the

receiving screen It is only by building up a picture based on many

observations that we can produce a clear diffraction pattern

Wave Mechanics We can solve very simple problems in quantum

physics using the SWE This is sometimes called WAVE MECHANICS There are very few problems that can be solved

exactly Approximation methods have to be used The simplest problem that we can solve is that of a

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

potential well This problem has recently become significant as it

can be applied to laser diodes like the ones used in CD players

Wave functions The wave function of a free particle moving

along the x-axis is given by

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

We cannot, however, measure y, we can only measure |y|2, the probability density

kxAx

Ax sin2

sin