The term 2 ℜ e ( A ( S ) A ∗ ( T )) in equation (1.12) is what generates quantum interference mathematically. We shall see that in certain circumstances the violations of equation (1.3) that are caused by quantum interference are not detectable, so standard probability theory appears to be valid. How do we know that the principle (1.11), which has these extraordinary consequences, is true? The soundest answer is that it is a fundamental postulate of quantum mechanics, and that every time you look at a digital watch, or touch a computer keyboard, or listen to a CD player, or interact with any other electronic device that has been engineered with the help of quantum mechanics, you are testing and vindicating this theory. Our civilisation now quite simply depends on the validity of equation (1.11).
6 Chapter 1: Probability and probability amplitudes Figure 1.2 The probability distribu- tions of passing through each of the two closely spaced slits overlap. 1.2.1 Two-slit interference An imaginary experiment will clarify the physical implications of the prin- ciple and suggest how it might be tested experimentally. The apparatus consists of an electron gun, G, a screen with two narrow slits S 1 and S 2 , and a photographic plate P, which darkens when hit by an electron (see Figure 1.1). When an electron is emitted by G, it has an amplitude to pass through slit S 1 and then hit the screen at the point x . This amplitude will clearly depend on the point x , so we label it A 1 ( x ). Similarly, there is an amplitude A 2 ( x ) that the electron passed through S 2 before reaching the screen at x . Hence the probability that the electron arrives at x is P ( x ) = | A 1 ( x ) + A 2 ( x ) | 2 = | A 1 ( x ) | 2 + | A 2 ( x ) | 2 + 2 ℜ e ( A 1 ( x ) A ∗ 2 ( x )) . (1 . 13) | A 1 ( x ) | 2 is simply the probability that the electron reaches the plate after passing through S 1 . We expect this to be a roughly Gaussian distribution p 1 ( x ) that is centred on the value x 1 of x at which a straight line from G through the middle of S 1 hits the plate. | A 2 ( x ) | 2 should similarly be a roughly Gaussian function p 2 ( x ) centred on the intersection at x 2 of the screen and the straight line from G through the middle of S 2 . It is convenient to write A i = | A i | e i φ i = √ p i e i φ i , where φ i is the phase of the complex number A i . Then equation (1.13) can be written p ( x ) = p 1 ( x ) + p 2 ( x ) + I ( x ) , (1 . 14a) where the interference term I is I ( x ) = 2 radicalbig p 1 ( x ) p 2 ( x ) cos( φ 1 ( x ) − φ 2 ( x )) . (1 . 14b) Consider the behaviour of I ( x ) near the point that is equidistant from the slits. Then (see Figure 1.2) p 1 ≃ p 2 and the interference term is comparable in magnitude to p 1 + p 2 , and, by equations (1.14), the probability of an electron arriving at x will oscillate between ∼ 2 p 1 and 0 depending on the value of the phase difference φ 1 ( x ) − φ 2 ( x ). In § 2.3.4 we shall show that the phases φ i ( x ) are approximately linear functions of x , so after many electrons have been fired from G to P in succession, the blackening of P at x , which will be roughly proportional to the number of electrons that have arrived at x , will show a sinusoidal pattern.
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