5.6: Summarizing this Wave/Particle Mess

What if We "Peek"?

In an attempt to solve the puzzle of the double slit from the previous section, we might try to simply watch the electrons as they pass through the slits. To do this, let’s put a bright light source between the slits, so that when electrons pass, by the light scatters off them and we see a small flash of light coming from the location of the electron, thereby telling us which slit it went through.

Figure 3.5.2 – Watching the Electrons as They Pass Through the Slits

When we do this, we find we have a problem.  The interference pattern disappears, and the previous “expected” pattern of electrons landing either opposite one slit or the other emerges.  Apparently we have affected the motion of the electrons after they pass through the slit with our detection device.  But of course we have!  Light has momentum, and when we scatter it off the electrons, the momenta of the electrons are altered, apparently ruining the interference effect we were trying to study.  So the obvious solution?  Use light with less momentum, so that it doesn’t transfer so much to the electrons.  The means we have to use light of long wavelengths.

Suppose we have a double slit with an unknown separation and want to determine that separation (we performed this exact experiment in 9B lab!).  The way to do it is to send some light through it, observe the interference, and do some math.  We have the relation between wavelength \(lambda\), slit separation \(d\), and angle \(\theta\)that the bright fringe is deflected to:

\[d\sin\theta_m=m\lambda\]

We can solve this for \(d\), and voilà, we know how far apart the slits are.  What happens if we use light with a very long wavelength?  Well, the quantity \(\sin\theta_m\) can never be larger than 1, so for a sufficiently long wavelength of light (namely, longer than \(d\)), there is no interference pattern (no dark fringes at all, just one bright fringe at the center).  That is, there are no \(\theta's\) to plug in.  We can't resolve the two slits – we can't draw any conclusions about their separation, or even tell if there are two slits!  As a general rule of thumb, we can say that in order to resolve two positions like the positions of these two slits, we need to use a wave that has a wavelength shorter than the dimension we are trying to resolve.  So if we want to watch the electrons so that we can see which slit they go through, we need to use light with a wavelength smaller than the slit separation.

Well it turns out that in order to stop affecting the motions of the electrons by using longer-wavelength light, the wavelength must be longer than the space between the slits. So we can use light that doesn't transfer enough momentum to significantly affect the trajectories of the electrons (keeping the interference pattern), but in doing so, the loss of resolution for that longer wavelength makes it impossible for us to determine which slit the electron goes through, which was our whole reason for introducing the light in the first place!  Infuriating... and amazing.

Quantum States

One thing this result tells us is that our classical notion of predicting the exact motion of particles using Newton's laws and kinematics must be discarded. Trajectories of particles are inherently probabilistic, and yet there is nevertheless a certain degree of predictability – the interference pattern is quite repeatable. So our task in studying this subject is to determine what physical properties contribute to the observed behavior, and come up with a mathematical model to predict – in a probabilistic manner – the results of experiments with these particles.

We already have an example of how we can use physical properties to predict this probabilistic behavior. The momentum of a particle is directly related to the associated wave's wavelength. So we can make a prediction of where no particles will land on the screen (dark fringes) by knowing how fast the particles were moving (and of course their masses) when we shot them at the double slit.

Given the weirdness of these quanta, it's not clear what all the properties are that define the probabilities we seek to predict. When we studied thermodynamics in Physics 9B, we defined something we called a "thermodynamic state". This was the equilibrium condition of a system, which was completely defined by several variables, like temperature, pressure, and volume. We will now define what we call a quantum state. Unlike a thermodynamic state, where knowing enough thermodynamic quantities to define the state tells us the values of other quantities (e.g. knowing the number of moles, volume, and pressure of an ideal gas tells us exactly its temperature), in quantum physics, we can usually only know probabilities of the state's values being measured.

We have greatly simplified things here (for example, we have not made any mention of how the quantum state evolves through time), but the general idea is this: We introduce something called a wave function \(\psi\left(x\right)\), which has the following properties:

• carries information about the probability of the particle being measured ("make a dot on a screen") at position \(x\),
• obeys the superposition principle, which means it can interfere with itself to create things like double-slit patterns and standing waves,
• has wave properties like wavelength (or a combination of wavelengths, if it is a superposition of waves) that correspond to physical properties like momentum,
• depends upon the physical situation that the particle finds itself in (like being acted-upon by external forces)

The topic of quantum mechanics is the study of solving for this wave function in various situations, and using it to make probabilistic predictions of what will be observed.