Quantum Mechanics

Anatomy of a classical travelling wave

adding varying amounts of an infinite number of waves Given the Uncertainty Principle, how do you write an equation of motion for a particle? First, remember that a particle is only a particle sort of, and a wave sort of, and it’s not quite like anything you’ve encountered in classical physics. We need to use Fourier’s Theorem to represent the particle as the superposition of many waves. adding varying amounts of an infinite number of waves sinusoidal expression for harmonics wavefunction of the electron amplitude of wave with wavenumber k=2p/l

probability for an electron to be found between x and x+dx Y(x,t) We saw a hint of probabilistic behavior in the double slit experiment. Maybe that is a clue about how to describe the motion of a “particle” or “wavicle” or whatever. We can’t write a deterministic equation of motion as in Newtonian Mechanics, however, we know that a large number of events will behave in a statistically predictable way. dx probability for an electron to be found between x and x+dx Y(x,t) 2 Assuming this particle exists, at any given time it must be somewhere. This requirement can be expressed mathematically as: you will find your particle once, not twice (that would be two particles) but once. If you search from hither to yon

Wavefunction requirements

Superposition Principle Conceptual definition When two or more wave moving through the same region of space, waves will superimpose and produce a well defined combined effect. Mathematical definition For a linear homogeneous ordinary differential equation, if        and        are solutions, then so is               . Standing Waves When two waves of equal amplitude and frequency but opposite directions of travel superimpose, you get a standing wave- a wave that appears not to move-its nodes and anti-nodes stay in the same place. This happens when traveling waves on a guitar string get to the end of the string and are reflected back.

The wavefunction is complex! Let’s try a typical classical wavefunction: For a particle propagating in the +x direction. Similarly, for a particle propagating in the –x direction: We also know that if Y1 and Y2 are both allowed waves, then Y1+Y2 must also be allowed (this is called the superposition principle). Oops, the particle vanishes at integer multiples of p/2, 2p/3, etc. and we know our particle is somewhere.

The wavefunction Can check that now the missing particle Graphical representation of a complex number z as a point in the complex plane. The horizontal and vertical Cartesian components give the real and imaginary parts of z respectively. Can check that now the missing particle problem is gone…. Note that we can construct a wavefunction only if the momentum is not precisely defined. A plane wave is unrealistic since it is not normalizable. You can’t get around the uncertainty principle!

Propogation of a free particle If you know the position of a particle at time t=0, and you constructed a localized wave packet, superimposing waves of different momenta, then the wave will disperse because, by definition, waves of different momenta travel at different speeds. the position of the real part of the wave… time Can solve propagation of an exponential (Gaussian wave-packet)? De Broglie waves are e^{I (k x-omega t)} – make superpositions to make a wave-packet….look at dynamics of wve-packet the probability density

Schrodinger Equation for a free particle What is a wave-equation that describes all these wave-functions? (the wave-equation would be k and omega independent) The Schrodinger Equation: Describes the time evolution of your wavefunction. Takes the place of Newton’s laws and conserves energy of the system. Since “particles” aren’t particles but wavicles, it won’t give us a precise position of an individual particle, but due to the statistical nature of things, it will precisely describe the distribution of a large number of particles. Superposition principle applies to the wave-equation (as for any wave-equation) Schrodinger saw de Broglie waves and said that all waves have a wave-equation – what is the wave-equation for a de Broglie wave?

Particle in a box Remember our guitar string? We had the boundary condition that the ends of the string were fixed. Quantum mechanical version- the particle is confined by an infinite potential on either side. The boundary condition- the probability of finding the particle outside of the box is ZERO!

The "Standing Waves" of Quantum Mechanics For the quantum mechanical case: For the guitar string: Make a superposition of incident and reflected Here show that time dependent part goes away when you multiply by the complex conjugate. Introduce the time independent schrodinger equation.

What the Uncertainty Principle tells us about the ground state Note that while Y=0 solves the problem mathematically, it does not satisfy the conditions for the quantum mechanical wavefunction, because we know that at any given time, the particle must be found SOMEWHERE. N=0 is not a solution. Appealing to the uncertainty principle can give us a clue.

Why quantization of angular momentum? an integer number of wavelengths fits into the circular orbit where l is the de Broglie wavelength We are beginning to understand...

The time-independent Schrodinger equation The wavefunction can be written as the sum of two parts. Note that: For a “stationary state” the probability of finding a particle is static!

Schrodinger Equation in the presence of forces expression for kinetic energy kinetic plus potential energy gives the total energy the potential Spring analogy