Young's double slit experiment - Quantum mechanical behaviour

Young's double slit experiment represents the observation of an interference pattern consistent with a wave nature for objects that traverse the apparatus. This is emphasised in the figures 3 and 4

Figure 3: Young's double slit experiment with water waves.

Figure 4: Young's double slit experiment, performed with either light or electrons leads to an interference pattern.

The diffraction and interference effects appear at first sight to be due to the beam of electrons, interfering with each other. However, the interference pattern still results even if only one electron traverses the apparatus at a time. In this case, the pattern is built up gradually from the statistically correlated impacts on many electrons arriving independently at the detection system. This effect is evidenced in figure 5.

Figure 5: Young's double slit experiment, performed with electrons in such a way that only one electron is present in the apparatus at any one time.

We see that the electron must in some sense pass through both slits at once and then interfere with itself as it travels towards the detector. Young's double slit experiment has been performed many times in many different ways with electrons (and other particles). The inescapable conclusion is that each electron must be delocalised in both time and space over the apparatus.

Considering the analogies between Young's double slit experiment performed with water waves, electro-magnetic waves and with electrons, and considering the material of the foregoing section, we can now specify some properties for a new theory of mechanics, termed wave mechanics.

1. There must be a wave function $\Psi(\mbox{\boldmath$r$} ,t)$ describing some fundamental property of matter. (We leave further physical interpretation of the wave function open to continuous debate through-out the course.)
2. As with the intensity pattern on the screen for water waves and light waves, the ``observable'' associated with the wave function indicating the probability of detection of the particle will be the intensity (square of the amplitude) of the wave. Mathematically, this is $\vert\Psi(\mbox{\boldmath$r$} ,t)\vert^2 = \Psi(\mbox{\boldmath$r$} ,t)\Psi^*(\mbox{\boldmath$r$} ,t)$
3. The wavelength in the wave function will be related to the de Broglie wavelength of the particle $\lambda = h/p$.
4. We would like to be able to proceed to develop a differential equation which would specify the time evolution of the wave function, consistent with the conservation of energy and momentum of physical systems.
5. The quantisation of energy should arise in a natural way from this formalism, just as it does for other bounded systems that support oscillations.
6. Then we must develop the formalism to enable other observables than simple the probability of detection ``position of the particle'' to be determined. Examples would be the energy and momentum of the particle.
7. Note the judicious use of the word observable. The actual wave function itself has never yet been observed.

It is clear that an improved understanding of waves in physics is now necessary. To this end, some results from wave motion in physics are reviewed.

A transverse wave train, travelling on a string in the $+x$-direction (as in figure 6) may be represented by

$$y(x,t)=A\cos 2\pi\nu\left(t-\frac{x}{v_p} \right)$$ (8)

where $\nu$ is the frequency of the wave and $v_p$ is its phase velocity. The phase velocity is the velocity with which a point on the wave maintaining the same phase appears to be transported.

$$v_p = \lambda \nu$$ (9)

Figure 6: A transverse wave train, travelling on a string in the $+x$-direction.

It is more common to define the angular frequency (frequency in radians/sec rather than cycles/sec)

$$\omega = 2\pi \nu$$ (10)

and the wavenumber

$$k = \frac{2\pi}{\lambda} \qquad \mbox{(by definition)}$$

$$= \frac{\omega}{v_p} \qquad \mbox{(substitution with the last two equations)}$$

$$= \vert p\vert/\hbar \qquad \mbox{(using de Broglie's relation)}$$ (11)

where $\hbar = h / 2\pi$. The wave equation for the wave moving in the $+x$-direction can now be written :

$$y(x,t)=A\cos(\omega t-kx)$$ (12)

In three dimensions, this equation would be

$$y({\bf r},t)=A\cos(\omega t-{\bf k.r})$$ (13)

It turns out that in quantum mechanics, a particle will be described as a wave packet. By this, we mean a group of (usually infinitely many) waves which mutually interfere, creating a new wave form which exhibits some localisation.

Figure 7: Two waves of nearly equal wavenumber combined coherently.

This can be illustrated by considering only two waves, of nearly equal wavenumber ($k\pm\Delta k$), and combining them coherently as in figure 7. Clearly, performing this process with many more waves would achieve a better localisation of the wave packet, as illustrated in figure 8.

Figure 8: Localisation of a wave packet by combination of many waves.

We find that

$$y = y_1 + y_2$$

$$= A\cos[(\omega +\Delta \omega /2)t-(k\Delta k /2)x] + A\cos[(\omega -\Delta \omega/2)t-(k-\Delta k/2)x]$$

$$.... \qquad \mbox{use trigonometric double angle formulae}$$

$$= 2A\cos(\omega t-kx)\cos(\frac{\Delta \omega}{2}t-\frac{\Delta k}{2}x)$$ (14)

The combined wave train exhibits the phenomenon of ``beats'' as shown in figure 7 where an amplitude modulation envelope is superimposed no the original wave train. The amplitude modulation envelope will clearly have the frequency $\Delta w$, wavenumber $\Delta k$ and hence the velocity $v_g = \Delta w / \Delta k$.

Exercise 2.3
Confirm the derivation of the combined wave by filling in the missing steps above.

The velocity of the localised group of waves (or beat) is known as the group velocity.

$$v_g = \frac{d\omega}{dk}$$ (15)

This must be compared to the phase velocity of each wave train making up the wave packet

$$v_p = \frac{\omega}{k}$$ (16)

Exercise 2.4
Show the group and phase velocities of a de Broglie wave for a relativistic particle are given by

$$v_g = v \qquad \mbox{and} \qquad v_p = c^2/v$$ (17)

Thus the de Broglie wave group associated with a moving particle travels with the same velocity as the particle. The de Broglie waves in the packet have superluminal velocities, however, these do not represent the motion of the particle, and therefore the special relativity is not violated.

Finally, the form of the wave equation, yielding the above expression for for a wave train is

$$\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$$ (18)

Exercise 2.5
Verify that $y(x,t)=A\cos(\omega t-kx)$ is indeed a solution of the wave equation. Clearly $y(x,t)=-iA\sin(\omega t-kx)$ is also a solution of the wave equation. It follows that

$$y(x,t)=Ae^{-i(\omega t-kx)}$$ (19)

is also a solution of the wave equation. This can be verified by direct substitution, or by exploiting the fact that any linear combination of solutions of the wave equation is itself a solution of the wave equation
(Hint: $e^{-i\theta} = \cos\theta -i\sin\theta$.)

Exercise 2.6
In fact a second order differentaial equation should have two constants of integration, which are determined by the boundary conditions of the specific problem. Show that for the equation above, we could write

$$y(x,t)=A\cos(\omega t-kx) + B\sin(\omega t-kx)$$ (20)

or

$$y(x,t)=Ce^{-i(\omega t-kx)} + De^{+i(\omega t-kx)}.$$ (21)

Also find the relationship between the two sets of coefficients. We will use the former set when discussing standing waves (like a guitar string), and the latter set when discussing travelling waves (like a ripple on a large pond). Make sure you appreciate this point.