Download Modern Physics

yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Elementary particle wikipedia, lookup

Molecular Hamiltonian wikipedia, lookup

Bell's theorem wikipedia, lookup

Quantum teleportation wikipedia, lookup

Density matrix wikipedia, lookup

Quantum entanglement wikipedia, lookup

Many-worlds interpretation wikipedia, lookup

Wheeler's delayed choice experiment wikipedia, lookup

History of quantum field theory wikipedia, lookup

Quantum electrodynamics wikipedia, lookup

Propagator wikipedia, lookup

Atomic theory wikipedia, lookup

Coherent states wikipedia, lookup

Identical particles wikipedia, lookup

Ensemble interpretation wikipedia, lookup

Max Born wikipedia, lookup

Renormalization group wikipedia, lookup

Measurement in quantum mechanics wikipedia, lookup

Erwin Schrödinger wikipedia, lookup

Canonical quantization wikipedia, lookup

Quantum state wikipedia, lookup

Hydrogen atom wikipedia, lookup

Path integral formulation wikipedia, lookup

T-symmetry wikipedia, lookup

Symmetry in quantum mechanics wikipedia, lookup

EPR paradox wikipedia, lookup

Dirac equation wikipedia, lookup

Interpretations of quantum mechanics wikipedia, lookup

Particle in a box wikipedia, lookup

Schrödinger equation wikipedia, lookup

Hidden variable theory wikipedia, lookup

Bohr–Einstein debates wikipedia, lookup

Copenhagen interpretation wikipedia, lookup

Double-slit experiment wikipedia, lookup

Probability amplitude wikipedia, lookup

Relativistic quantum mechanics wikipedia, lookup

Wave function wikipedia, lookup

Wave–particle duality wikipedia, lookup

Matter wave wikipedia, lookup

Theoretical and experimental justification for the Schrödinger equation wikipedia, lookup

Modern Physics
lecture 3
Louis de Broglie
1892 - 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,
E hf h
c 
Which says that the wavelength of light is related to its
Making the same comparison for matter we find…
p mv
Quantum mechanics
Wave-particle duality
Waves and particles have interchangeable properties
This is an example of a system with complementary
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
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
DxDpx   / 2
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
 y 1  y
 2 2
v t
Erwin Schrödinger
1887 - 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 y 1 2 y
 2 2
v t
• By making this substitution we find that
 2ψ
 2 ψ
• Where w /v = 2p/
• Thus
w 2/ v 2  (2p/)2
p = h/
Energy and the Schrödinger Equation
Consider the total energy
Total energy E = Kinetic energy + Potential Energy
E = m v 2/2 +U
E = p 2/(2m ) +U
 Reorganise equation to give
p 2 = 2 m (E - U )
ω 2 2m
 2 E  U 
 From equation on previous slide we get
• Going back to the wave equation we have
 2ψ 2m
 2  E  U ψ  0
• 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
  2 ( E  U )y
The wave function contains all of the information
that can be known about a particle
Solution to the SWE
The solutions y(x) are called the STATIONARY
STATES of the system
The equation is solved by imposing BOUNDARY
The imposition of these conditions leads naturally
to energy levels
If we set U  
 4πε  r
0 
We get the same results as Bohr for the energy levels of the
one electron atom
The 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
y  y *y
This leads to an interpretation of the double slit
Interpretation of the Wavefunction
Max Born suggested that y was the PROBABILITY
AMPLITUDE of finding the particle per unit volume
|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
Since the chance of finding the particle somewhere in
space is unity we have
 ψ ψ* dV   ψ
dV  1
• When this condition is satisfied we say that the wavefunction
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
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
 2px 
y x   A sin 
  A sin kx
  
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