Modern Physics lecture IV

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