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Modern Physics lecture IV

Modern Physics lecture IV. 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,.

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Modern Physics lecture IV

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  1. Modern Physicslecture IV

  2. Louis de Broglie1892 - 1987

  3. 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…

  4. 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”

  5. The Uncertainty Principle Measurement disturbes the system

  6. 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

  7. 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/2p) • Energy / Time (DEDt > h/2p) • Amplitude / Phase

  8. 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

  9. Erwin Schrödinger1887 - 1961

  10. Solution to the Wave equation • We consider a trial solution by substituting y(x,t) = y(x) sin(wt) into the wave equation • By making this substitution we find that • Wherew/v =2p/landp = h/l • Thus • w2/v2 = (2p/l)2

  11. Energy and the Schrödinger Equation • Consider the total energy Total energy E = Kinetic energy + Potential Energy E = mv2/2+U E = p2/(2m)+U • Reorganise equation to give p2=2m(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

  12. 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

  13. 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

  14. 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

  15. Interpretation of the Wavefunction • Max Born suggested that y was the PROBABILITY AMPLITUDE of finding the particle per unit volume • Thus |y|2dV =yy*dV (y*designates complex conjugate)is the probability of finding the particle within the volume dV • The quantity |y|2is 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

  16. Max Born

  17. 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

  18. 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

  19. 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

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