Modern Physics lecture 1

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Modern Physicslecture 1
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Louis de Broglie1892 - 1987
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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,

• Which says that the wavelength of light is
  related to its momentum
• Making the same comparison for matter we find

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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

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The Uncertainty Principle
Measurement disturbes the system
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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 Heisenburg
  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

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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 gt h/4p)
• Energy / Time (DEDt gt h/4p)
• Amplitude / Phase

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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

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Erwin Schrödinger1887 - 1961
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Solution to the Wave equation

• We consider a trial solution by substituting
• y (x, t ) y (x ) sin(w t )
• into the wave equation

• By making this substitution we find that

• Where w /v 2p/l and p h/l
• Thus
• w 2/ v 2 (2p/l)2

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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 )
• From equation on previous slide we get

• Going back to the wave equation we have

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

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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

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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

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
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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

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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

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

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Max Born
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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

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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

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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 y2, the probability density