Physics 361 Principles of Modern Physics Lecture 7

More de Broglie Matter Waves This lecture Complex numbers and representation of waves. Waves in motion Adding quantum amplitude waves to get wave packets. Next lecture More about wave packets Uncertainty Principle Wave packets in motion Schrodinger equation

Rule of wave (quantum) mechanics This permits interference (wave- like) effects to be observed. Indistinguishable This does not allow for interference effects – we have in this case, classical particle-like behavior. Distinguishable Actual Path Indistinguishable (not determined) paths or events have their wave functions added. The resulting “quantum intensity”, which is proportional to the probability per unit length of observing a particle is: Distinguishable (determined) paths or events have the “intensity” of the wave functions added.

How do we represent these Quantum Mechanical waves? While one often sketches waves in Quantum Mechanics just like classical waves – there are some important differences!!

Let’s first consider classical waves (Like waves on a string, sound waves, or low-amplitude waves in water) Consider a boat out at see riding waves. If we take a snapshot of the boat on the wave, can we determine if it is just about to go over a crest, or whether instead it just went over a crest?

How do we determine direction of motion for classical waves? Consider a boat out at see riding waves. If we take a snapshot of the boat on the wave, can we determine if it is just about to go over a crest, or whether instead it just went over a crest? No, we need to know the direction the wave is travelling!!

For classical waves, we need initial amplitude and velocity to determine wave. A velocity measurement of the wave at any point in time will give us all the information we need to know its direction, and thus to completely describe its motion.

For Q.M. waves, we only need the initial amplitude, not it’s velocity This is possible because there is another component of Q.M. waves which determines its direction. This extra component in the wave is conventionally represented by the imaginary axis of the complex plain. What one typically sketches, is just the real component.

Review of the complex plane and its use in describing classical waves and oscillatory behavior The complex plane is a 2D spacing consisting of a real (x) axis and an imaginary (y) axis Any vector in the complex plane represents a complex number

The oscillation or wave becomes a vector that rotates in the complex plane The imaginary axis just makes it easier to solve problems and visualize oscillations and waves. Wave number (or ‘wave vector’ in 1D)

The oscillation or wave becomes a vector that rotates in the complex plane The imaginary axis just makes it easier to solve problems and visualize oscillations and waves. Wave number (or ‘wave vector’ in 1D)

The real projection is The imaginary projection is Wave motion A positive going wave can be represented by subtracting the phase.

For classical waves we just keep the real portion This is the projection of the vector along the real axis. The real projection is The imaginary projection is

Adding waves is like adding vectors At a specific time t and position x, the contribution from each wave is added. The real projection is The imaginary projection is

Adding waves is like adding vectors At a specific time t and position x, the contribution from each wave is added. total vector The real projection is The imaginary projection is

We can add up a bunch of waves at a specific time to obtain a new real amplitude (projection along Re axis) The real projection of the classical wave summation would look something like this. Notice that the summation of the top three waves yields a total wave that “beats” in space. (We will revisit this effect when we discuss the uncertainty principle and wave packets).

Now back to Q.M. waves For Q.M. waves, the wave function has two components, Re and Im. The probability per unit length for finding the particle is (as discussed earlier)

Now back to Q.M. waves For Q.M. waves, the wave function has two components, Re and Im. This has real consequences! For example -- The probability per unit length for finding the particle is (as discussed earlier): A classical wave which undulates will now give a constant probability! To calculate, multiply by its complex conjugate.

Calculating Probabilities Complex conjugate takes all the Im terms (regardless of where they are) and changes their sign. For example, So (Pythagorean theorem) for a wave with a single wave vector. Notice that this is a constant! -So the physically relevant quantity in Q.M. for a single wave does not depend on its spatial position. Only in interference do we detect the wave nature in Q.M. Q.M. Classical

Euler’s formula for representing a complex value Euler’s formula is a famous relation between exponentials and sinusoidal functions in the complex plane. It can be derived through comparing terms of the Taylor expansions (though we won’t go through that in this course). If we represent a wave function with Euler’s formula, we need to know how to take the complex conjugate. This is the same as before!! Change the signs of all terms that are imaginary. Thus, So, as before.

So we have two components for Q.M. waves. How does this allow us to determine the waves direction? Let’s use Euler’s formula to represent the QM wave, There are two ways of mathematically representing a forward going (+x direction) wave. These have differences in the exponents. Negative going waves have the terms in exponent summed. These can be worked out from the displacement of the wave according to the complex wave vector.

Choose a description of waves and see how this allows us to determine a wave’s direction from a snap shot of the wave. We will use This will allow the direction to be specified by the k vector only!! (Note: this is only a convention, and it is not always chosen like this.)

How to determine the waves direction from a snap shot of wave. The relative phases along the x-axis of the Imaginary components with respect to the Real components determines the waves direction. Re Im Re Im Lags by Leads by