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R. Barrett and P. P. Delsanto
so-called atoms of electricity, should also display wave-like characteristics,
such as diffraction and interference, which we discussed in Chap. 1 for light.
This prediction was confirmed by the experiments of G. P. Thomson and of
Davisson and Germer (1923–1927), who diffracted a beam of electrons from
a nickel crystal.
But how can we depict a particle with a dual nature? Classically we describe
a particle’s location by specifying its coordinates in four dimensions (4D)—
the three spatial dimensions and time. In the case of wave motion, we
normally specify the amplitude of the wave in terms of these 4D coordinates.
The physical interpretation of amplitude depends on the type of wave we are
considering and of the medium supporting it. For instance, in the case of a
sound wave, amplitude represents the displacement of the medium through
which the wave is propagating (usually air). In the case of light and other
forms of electromagnetic radiation propagating through a vacuum, there is no
physical supporting medium, and the amplitude represents the value of the
electric (or magnetic) field at these 4D coordinate points, bearing in mind
that the amplitude may be either positive or negative. The intensity of the
light is given by the square of the amplitude, ignoring the sign.
Now we see the conundrum that presented itself to physicists in the early
twentieth century. Common Sense will get us nowhere. Instead we must
abandon ourselves to mathematics, and see where it leads. The modern interpretation is that associated intimately with every particle is a wave function.
This wave function is a mathematical abstraction, and as far as we know
has no physical reality. Indeed, it is a complex quantity, which in mathematics means that it has both real and imaginary parts. (As we have seen in
Sect. 4, an imaginary number comes from taking the square root of a negative number.) If the reader is getting the sinking feeling that we are about to
lose ourselves in an unintelligible quagmire, please take heart, for we shall go
no further in this direction.
Clearly the wave function must have some connection with reality, or it
would remain an irrelevant mathematical oddity. The connection comes via
its modulus squared . 5 Not just electrons, but all matter, has wave-like properties. However, the interference effects are easier to observe with less massive
particles. In analogy with the intensity of light described above, the modulus
squared of the wave function at a particular point in space and time gives
the probability of finding the particle to be located at that point at that time.
5 The modulus (or absolute value) of a real quantity is its magnitude, ignoring whether it is positive
or negative. However, wave functions are complex quantities, which complicates matters slightly. The
modulus squared of a complex quantity is the sum of the squares of the real and imaginary parts.
R. Barrett and P. P. Delsanto
so-called atoms of electricity, should also display wave-like characteristics,
such as diffraction and interference, which we discussed in Chap. 1 for light.
This prediction was confirmed by the experiments of G. P. Thomson and of
Davisson and Germer (1923–1927), who diffracted a beam of electrons from
a nickel crystal.
But how can we depict a particle with a dual nature? Classically we describe
a particle’s location by specifying its coordinates in four dimensions (4D)—
the three spatial dimensions and time. In the case of wave motion, we
normally specify the amplitude of the wave in terms of these 4D coordinates.
The physical interpretation of amplitude depends on the type of wave we are
considering and of the medium supporting it. For instance, in the case of a
sound wave, amplitude represents the displacement of the medium through
which the wave is propagating (usually air). In the case of light and other
forms of electromagnetic radiation propagating through a vacuum, there is no
physical supporting medium, and the amplitude represents the value of the
electric (or magnetic) field at these 4D coordinate points, bearing in mind
that the amplitude may be either positive or negative. The intensity of the
light is given by the square of the amplitude, ignoring the sign.
Now we see the conundrum that presented itself to physicists in the early
twentieth century. Common Sense will get us nowhere. Instead we must
abandon ourselves to mathematics, and see where it leads. The modern interpretation is that associated intimately with every particle is a wave function.
This wave function is a mathematical abstraction, and as far as we know
has no physical reality. Indeed, it is a complex quantity, which in mathematics means that it has both real and imaginary parts. (As we have seen in
Sect. 4, an imaginary number comes from taking the square root of a negative number.) If the reader is getting the sinking feeling that we are about to
lose ourselves in an unintelligible quagmire, please take heart, for we shall go
no further in this direction.
Clearly the wave function must have some connection with reality, or it
would remain an irrelevant mathematical oddity. The connection comes via
its modulus squared . 5 Not just electrons, but all matter, has wave-like properties. However, the interference effects are easier to observe with less massive
particles. In analogy with the intensity of light described above, the modulus
squared of the wave function at a particular point in space and time gives
the probability of finding the particle to be located at that point at that time.
5 The modulus (or absolute value) of a real quantity is its magnitude, ignoring whether it is positive
or negative. However, wave functions are complex quantities, which complicates matters slightly. The
modulus squared of a complex quantity is the sum of the squares of the real and imaginary parts.