Physics 361Principles of Modern Physics

Lecture 8

More de Broglie Matter Waves

This lecture• More about wave packets• Uncertainty PrincipleNext few lectures• Wave packets in motion• Schrodinger equation• Solving Problems with Schrodinger equation

Understanding Interference Through (Complex) Vector Representation

At a specific time t and position x, the contribution from each wave is added like a vector in the complex plane.

totalvector

Understanding Interference Through (Complex) Vector Representation – double-slit exampleIn the case where interference is observed no significant amount of energy is lost as the particle goes through the slits. Therefore, the magnitude of momentum, energy, and wavelength remain constant.

If the path length difference is a half wavelength plus some other integer multiple of wavelengths then the vectors point in opposite directions in complex plane

Understanding Interference Through (Complex) Vector Representation – double-slit exampleSince energy remains constant, the rate that the vectors rotate around in the complex plain is the same. This frequency is given by:

As the vectors rotate together, they always cancel. So the detector always measures a local minimum in electron counts.

Understanding Interference Through (Complex) Vector Representation – double-slit exampleSince energy remains constant, the rate that the vectors rotate around in the complex plain is the same. This frequency is given by:

As the vectors rotate together, they always cancel. So the detector always measures a local minimum in electron counts.

Understanding Interference Through (Complex) Vector Representation – double-slit example

For wave vectors in the same direction, they rotate together always pointing in the same direction.

Understanding Interference Through (Complex) Vector Representation – double-slit example

For wave vectors in the same direction, they rotate together always pointing in the same direction.

Understanding Interference Through (Complex) Vector Representation – double-slit example

For wave vectors in the same direction, they rotate together always pointing in the same direction.

So we always have a local maximum number of electron counts detected!

Wave Packets

Path determined

Path not determined

Before we argued that we could use light to resolve a particles position with a precision given by approximately the wavelength of the light.

Once the particle’s position is determined, it must then continue on as a wave, but how can you specifiy the position on a wave?

Wave PacketsWe can make waves become clumpy by adding up waves of different frequencies.

At the right is an example from before where we added waves with three different wave vectors.

Notice that the sum results in probability magnitudes which are clumpy.

However, a major problem is that the clumps repeat forever (ie, to infinity)!! We have not isolated the particle.

Let’s use the vector representation of waves to understand this effect.

Let’s fix time and only vary the position in space along the x-axis and plot the vectors for several waves

Let’s assume each wave has a vector given by

These will have equal angles between them when represented on the complex plane for a specific x position.

Let’s fix time and only vary the position in space along the x-axis and plot the vectors for several waves

Let’s assume each wave has a vector given by

As the position x changes, their phase differences will vary.

As they become aligned we approach a wave packet.

Let’s fix time and only vary the position in space along the x-axis and plot the vectors for several waves

When each wave has the same angle, we obtain a maximum amplitude of oscillation

How can we determine the condition for a maximum?

When are we in a wave packet?

Since the wave vectors are equally spaced such as,

let’s find when successive ones are aligned, this will be the same as when they all are. That is, when

We can ask how far between clumps. This is

So we expect an infinite number of wave packets.

They occur with a physical spacing given by

or

We can write the vector addition using Euler’s formula. We must sum over all the wave vectors

so that

We can make one wave packet if we us a continuous distribution of wave vectors.

If thenfrom

This means we have a continuous distribution of wave vectors.The discrete sum before

Now becomes an integral

where is a continuous variable.

Uncertainty Relationships

Let’s consider a range of wave vectors which are only finite within a small window

This can be constructed with a Gaussian distribution

If we put that into our integral for determining the wave sum we obtain

This is only large for

or less.

Uncertainty Relationship For Momentum

We thus can only know the product of the uncertainty in position and wave vector to no better than

Since

We can also write the uncertainty relationship as