Introduction.
When Max Planck (1858-1947) introduced the quantum of energy to explain
black body radiation and Albert Einstein (1879-1955) used the quantum to
explain the photoelectric effect, they were applying the particle or
atomic hypothesis to energy as it had been applied to matter. It was
Isaac Newton (1642-1727) who introduced the atomic hypothesis into Modern
Physics to explain the light phenomena. But his particle theory
of light was totally rejected by the physicists when the wave
nature of light was discovered by Thomas Young (1773-1829) and
Augustin Fresnel (1788-1827).
It was the chemists of the nineteenth century who applied the
atomic hypothesis to matter to explain chemical reactions. By
the end of the century after much controversy the atomic hypothesis
of matter was accepted by all chemists. As the evidence for the
discrete structure of matter was being found by the chemists,
the discrete structure of electricity was discovered by the physicists.
With the discovery of the electron by J. J. Thompson (1856-1940) in 1897,
the atomic hypothesis was extended to electricity. After Ernest
Rutherford (1871-1937) performed the alpha particle scattering experiments
which established the nuclear model of the atom, Neils Bohr (1885-1962)
proposed in 1913 the planetary model of the atom, where the negative
charged electrons move in discrete orbits about the positively charge
nucleus of the atom. By quantizing the angular momentum of the orbiting
electrons, Bohr showed that the electrons occupied certain discrete orbits.
Bohr showed that this implied that the energy of the electrons in their
orbits is also quantizied in discrete energy levels and that the electrons
absorb or radiate the difference between the energy of the energy levels
of the orbits as the electron jumped from one energy level to another.
"A consideration of these problems [Bohr's theory of the atom] led me, in 1923, to the conviction that in the theory of Matter, as in the theory of radiation, it was essential to consider corpuscles and waves simultaneously, if it were desired to reach a single theory, permitting of the simultaneous interpretation of the properties of Light and of those of Matter. It then becomes clear at once that, in order to predict the motion of particles, it was necessary to construct a new Mechanics -- a Wave Mechanics, as it is called today -- a theory closely related to that dealing with wave phenomena, and one in which the motion of a corpuscle is inferred from the motion in space of a wave. In this way there will be, for example, light corpuscles, photons; but their motion will be connected with that of Fresnel's wave, and will provide an explanation of the light phenomena of interference and diffraction. Meanwhile it will no longer be possible to consider the material particles, electrons and protons, in isolation; it will, on the contrary, have to be assumed in each case that they are accompanied by a wave which is bound up with their own motion. I have even been able to state in advance the wavelength of the associated wave belonging to an electron having a give velocity." [1]In 1924 de Broglie proposed that a material particle such as an electron might have a dual nature. In the study of light, the wave properties involves wavelength λ, frequency f, and speed of propagation c of the wave. In terms of these properties, the quantum theory of light defines the mechanical properties of the light corpuscles as follows:
For the study of material particles, it is the other way around;
it starts with the mechanical properties of the particles of mass
m, momentum p, and energy E and these are
measurable quantities. So when de Broglie postulated that such
a particle may also have wave properties, he attempted to derive
their frequency f and wavelength λ of the associated wave
in terms of their mass m, momentum p, and energy E.
In so doing, de Broglie was guided by the relations for light
(equations
[2] and
[3]).
He argued that associated with a particle
is a wave of frequency f and wavelength λ given by
f = E/h,
and λ = h/p. [9]
De Broglie started with the simple case of an isolated particle,
that is, one separated from all external influences. With this
particle he associated a wave. Now consider a reference system 0
(x0, y0, z0,
t0) in which the particle is at rest. According
to the Theory of Relativity this is the "proper" system
for the particle. Within such a system the wave will be stationary,
since the particle is at rest; its phase will be the same at every point,
and it will be represented by an expression of the form
y = A cos 2πf(t - x/w),
where A is the amplitude (that is, the maximum displacement
on either side of the x-axis) of the wave moving with phase
velocity w along the x-axis whose vibrations are
in the y direction. In the proper system,
y = A cos 2πf0
(t0 - x0/w),
where t0 is the "proper" time, and
x0 is the "proper" location of the wave
in the direction of the x0-axis. According to the
principle of inertia, the particle will be moving with uniform
rectilinear motion in every Galilean system. Let us consider
such a Galilean system and let v be the velocity of the
particle in this system. Without loss of generality, let us take
the direction of motion of the particle to be along the x-axis.
According to the Lorentz transformations, the time t measured
by an observer in this new system of reference is connected to the
proper time t0 by the relation:
t0 = γ(t + vx/c2),
where γ = 1/α =
1/√(1- v2/c2) with
α2 = (1- v2/c2) and
α2 = 1 - v2/c2 =
1 - β2 with β = v/c.
Similarly the location x measured by an observer in the
new system of reference is connected to the proper location
x0 by the relation:
x0 = γ(x + vt).
Hence for such a observer the phase of the wave will be given by
y = A cos{(2πf0/α)[t +
vx/c2 - x/w - v/w]}.
Let w = c2/v or
1/w = v/c2, then
y = A cos(2πf0/α)
[t + x/w - x/w -
v2t/c2] =
A cos(2πf0/α)
[t - v2t/c2] =
A cos(2πf0/α)
[t(1 - v2/c2)] =
A cos(2πf0/α)[tα2] =
A cos(2παf0t) =
A cos 2πft,
where the wave will have a frequency
f = αf0
and will move along the x-axis with a phase velocity
w = c2/v. [10]
If we assume
equations [9]
and use the relativistic relation
E = mc2, it can be shown that the phase velocity
w, or velocity of propagation of the associated waves is
w = λf = (h/p)(E/h) =
E/p = mc2/mv =
c2/v,
where v < c for a material particle,
c2/v > c and
w > c. That is, the de Broglie
phase velocity w is always greater than the c.
But this is not a problem, because w is neither the velocity
v of a material particle nor, as will be shown later, the group
velocity of a set of waves. In fact, the wave associated with
a material particle are not to be regarded as being mechanical
or electromagnetic, but rather are "probability waves."
As we shall see later, this means that the intensity of waves
at any point will be taken as giving the fraction of a large number
of similar particles, emitted with the same velocity, that will
reach a given area in unit time, or the probability of one particle
reaching the area. The waves are thus a device for computing
the probability that a particle will behave in a certain way.
De Broglie argued that in the case of light one cannot think
of a unit of energy without associating with it a wavelength and
frequency, and that therefore material particles, like photons,
must be accompanied by phase velocity.
Let us now derive the wave properties for a particle in terms
of its energy and momentum. First, let us derive the frequency
of the wave in terms of the energy of the particle. According
to Einstein's mass-energy equivalence relationship,
E2 = p2c2 +
E02.
In the proper system of particles, where p = 0, the total energy
E of a particle is reduced to its internal energy E0;
that is,
E = E0 = m0c2,
since
E0 = m0c2
(where m0 is the proper or rest mass).
And since the quantum relation
E = hf (
equation [2])
holds in all proper systems, then
E0 =
hf0 =
m0c2. [11]
According to this relation, the proper frequency f0 is a
function of the proper mass m0 of the particle, and
inversely. Hence,
E = hf0.
Since this relation holds for all Galilean systems, then the frequency
of the wave associated with a particle having energy E is, since
E = hf,
f = E/h. [12]
Now, let us derive the wavelength of the wave in terms of the
momentum of the particle. According the Special Theory of Relativity,
the momentum p of a particle is
p = γm0v =
mv = mc2v/c2 =
Ev/c2,
and using
equation [10],
v = c2/w,
p = (E/c2)(c2/w) =
E/w = hf/λf = h/λ.
Hence,
p = h/λ, [13]
or
λ = h/p = h/mv, [9b]
where λ is the distance between two successive wave crests.
This is a fundamental relation of de Broglie quantum theory of matter
and is called the de Broglie wavelength.
Davisson and Germer were studying the scattering of electrons from a solid using an apparatus in which a beam of electrons was bounced off various target substances to study the angles at which they are reflected. The energy of the electrons in the primary beam, the angle at which they are incident upon the target, and the position of the detector can all be varied. Classical physics predicted that the scattered electrons will emerge in all directions with only a moderate dependence of their intensity upon scattering angle and even less upon the energy of the primary electrons. Using a block of nickel as the target, Davisson and Gremer verified these predictions. But in the course of their work there occurred an accident that allowed air to enter the apparatus and oxidize the nickel surface. To reduce the oxide to pure nickel, the target was baked in a high temperature oven. After treatment, the target was returned to the apparatus and the measurements were resumed. Now the results were very different from what had been found before the accident. They had never heard of electron diffraction and were puzzled to find that at certain specific angles, the reflected electron beams were intense, but dropped off almost to zero at other angles. Instead of a continuous variation of scattered electron intensity with angle, a distinct maxima and minima were observed whose position depended upon the electron energy. In 1926, they presented some of their data to an international conference of physicists without an explanation. It was suggested to them that possibly their results were an example of electron interference. They at once began checking their data against de Broglie's equation for the electron wavelength, and found almost perfect agreement. It happened that the electrons accelerated by the usual apparatus attain velocities that make their de Broglie wavelength close to those of X-rays. Thus the crystals reflecting beams of electrons give off electron diffraction patterns similar to those of X-rays.
De Broglie's hypothesis suggested the interpretation that the electron waves were being diffracted by the target, much as X-rays are diffracted by Bragg reflection from crystal planes. This interpretation received support when it was realized that the effect of heating a block of nickel at high temperature is to cause the many small individual crystals of which it is normally composed to form into a single large crystal, all of whose atoms are arranged in a regular lattice.
Let us verify that de Broglie waves are responsible for the results
of the Davison and Germer experiment. In a particular experiment,
a beam of 54-ev electrons was directed perpendicularly at the
nickel target, and a sharp maximum in the electron distribution
occurred at an angle of 50° to the original beam. The angles of
incidence and scattering relative to the family of Bragg planes
will both be 65°. The spacing of the planes in this family,
which can be measured by X-ray diffraction is 0.91 Angstroms (Å).
The Bragg equation for maxima in the diffraction pattern is
nλ = 2d sin θ where n = 1, 2, 3, ....
Since d = 0.91 Å and θ = 65 °, assuming that
n = 1, the de Broglie wavelength of the diffracted electrons is
λ = 2 × 0.91 Å × sin 65° = 1.65
Å.
To calculate the expected wavelength of the electrons, the electron
momentum must be determined from their kinetic energy
K = mv2/2. Since the kinetic energy of
54 ev is small compared with its rest energy
m0c2 of 5.1 × 105,
the relativistic considerations can be ignored.
The unit used to measure the kinetic energy of the particles is called the
electron-volt. One electron-volt (1 ev) is defined as the amount of
energy acquired by an electron as it falls through a potential difference of
1 volt. This is equivalent to 1.602 × 10-19 joules,
since the charge on the electron is 1.602 × 10-19 coulombs.
Thus
1ev = (1.602 × 10-19 coulombs) × (1 volt) =
1.602 × 10-19 joules,
since 1 volt is equal to 1 joule per coulomb. This unit is too small for
convenience in atomic physics, so an unit of one million electron volts (Mev)
is used and is equal to
1 Mev = 1.602 × 10-13 joules. Therefore,
mv = √[2mK] = √[2 ×
(9.1 × 10-31 kg)
(54 ev × 1.602 × 10-19 j/ev)]
mv = 4.0 × 10-24 kg-m/sec.
Using the de Broglie' wavelength,
λ = h/p = h/mv, [9b]
let us calculate the expected wavelength of the electrons,
λ = (6.63 × 10-34 j-sec)/(4.0 × 10-24
km-m/sec) = 1.66 × 10-10 m = 1.66 Å.
This gives excellent agreement with the observed wavelength.
The Davison-Germer experiment thus seem to provide direct verification
of de Broglie's hypothesis of the wave function associated with a particle.
But the experiment does not confirm the wave nature of moving bodies.
What is being diffracted here is not a lone single electron, but a whole beam
of electrons. The De Broglie wave does not represent an individual electron,
but a statistical aggregate of particles; the wave is not the nature of
electron, but is a "wave of probablity", a periodic function, indicating
the probability of an occurence of a particle at a certain place. This
statistical interpretation of De Broglie's waves, first proposed in 1926 by
Max Born and W. Heisenberg,
was gradually accepted by most, though by no mean all, physicists,
and has become an almost orthodox interpretation.
Electrons are not the only particles whose wave behavior can be demonstrated. The diffraction of neutrons and of whole atoms when scattered by suitable crystals has been observed, and in fact neutron diffraction, like X-ray and electron diffraction, is today a widely used tool for investigating crystal structures.
As in the case of electromagnetic waves, the wave and particle aspects of moving bodies can never be simultaneously observed, so that we cannot determine which is the "correct" description. All we can say is that in some respects a moving body exhibits wave properties and in other respects it exhibits particle properties. Which set of properties is most conspicuous depends upon how the de Broglie wavelength compares with the dimensions of the bodies involved: the 1.66 Angstrom (Å) wavelength of a 54-ev electron is of same order of magnitude as the lattice spacing in a nickel crystal, but the wavelength of an automobile moving at 60 mph is about 5 × 10-23 ft, far too small to manifest itself.
Schrodinger considered a particle of mass m with linear
momentum p and total energy E to be in a field of force
of potential V(x, y, z, t),
so that the (non-relativistic) equation of total energy is the
sum of kinetic energy K and potential energy V, or
E = K + V = mv2/2 + V.
Now let us express the kinetic energy K in terms of linear momentum
p = mv:
K = mv2/2 =
m2v2/2m =
p2/2m,
we get
E = p2/2m + V. [14]
This classical expression for the total energy of a particle is
called the Hamilton's equation. Now Schrodinger expressed
this energy equation as an operator equation:
Eψ = (p2/2m + V)ψ =
p2ψ/2m + Vψ. [15]
This is Schrodinger's operator equation and may be understood
in two ways.
First, as an observational statement, it states that any achievable
state ψ must be such that measurements of the total energy E
gives the same result as the sum of the measurement of the kinetic
energy, p2/2m, and of the potential energy V.
This ensures that energy is conserved.
Secondly, Schrodinger's equation states that the comparison of
the same state function at two successive instants of time must be
related to the comparison at two neighboring points in space in a very
definite and precise way, sufficiently precise to permit the actual
determination of it to be carried out. In other words, the equation
can be solved for ψ.
The problem that Schrodinger set himself to solve was: What kind
of de Broglie waves, which satisfy certain restrictions, can exist
permanently in a field of force surrounding the nucleus of the
atom? Schrodinger confined his attention, at first, to standing
waves, or vibrations. The wave distribution that he was seeking constituted
therefore the modes of vibration of the de Broglie waves in the field of force
about the nucleus of the atom. These modes represent the stable states
of the atom. To obtain the modes, let us start with the d'Alembert's
wave equation, which describes wave motion in general. Let
u(x, y, z, t)
be a function of x, y, z, and t, which defines,
at each point (x, y, z) of space and at each instant of
time t, the state of vibration of the wave disturbance. This function
u must satisfy the d'Alembert's wave equation.
∂²u/∂x² +
∂²u/∂y² +
∂²u/∂z² -
(1/w2)(∂²u/∂t²) =
0, [16]
where w is the velocity of propagation of the waves at
the point (x, y, z) at time t.
To simplify these equations we shall use symbol ∇²u
for the sum of the three second-order partial derivatives of the function
u with respect to the Cartesian coordinates
x, y, z. This is called the Laplacian operator
and is defined as
∇² =
∂²/∂x² +
∂²/∂y² +
∂²/∂z².
Using this convention, the wave equation [16] becomes
∇²u -
(1/w2)(∂²u/∂t²) =
0. [17]
Now in order to determine the standing sinusoidal waves that can
be formed, let us assume that function u represents a standing
wave which may be defined by
u = ψ(x, y,
z)cos(2πft), [18]
where f is the frequency of the standing waves and
ψ(x, y, z) defines the maximum swing,
or the amplitude, of the wave vibration at the point
(x, y, z). The function
ψ(x, y, z) is unknown but by substituting
equation [18] into equation [17] we get the following equation
that ψ must satisfy:
∇²ψ +
(4π2f2/w2)ψ =
0. [19]
Using the Hamilton's energy
equation [14]
above, the linear momentum of the particle p may be expressed in terms
of energy (E - V):
p = √[2m(E - V)]. [20]
If we associate with this moving particle a de Broglie wave with
a frequency f given by
E = hf,
and a de Broglie wavelength λ given by
λ = h/p,
then the phase velocity w of the de Broglie wave is
w = fλ = (E/h)(h/p) =
E/√[2m(E - V)]. [21]
Hence, substituting the equation [21] into equation [19],
we get Schrodinger's equation:
∇²ψ +
(8π2f2m/E2)
(E - V)ψ = 0, [22]
or, since E = hf, we get
∇²ψ +
(8π2m/h2)
(E - V)ψ = 0. [23]
The problem of the particle in the box is somewhat artificial, since there is no such a thing as an infinitely great potential energy, and a particle cannot be made completely free from all external influences while thus confined. Nevertheless, the problem is an important one, because it reveals the quantization of the energy. Energy quantization occurs, basically, because only certain discrete values of the wavelength can be fitted between its boundaries.
Schrodinger with his equation describes a particle by its wave function and he goes on to show how this particle wave function evolves in space and time under a specific set of circumstances. One such circumstances of great interest is that of a single electron moving in the electric field of a proton. Using his wave equation, Schrodinger was able to show that the electron wave function can assume only certain discrete energy levels, and that those energy levels are precisely the same as the energies of the electronic orbits of the hydrogen atom, postulated by Bohr. The particle wave function is a mathematical expression describing all the observable features of a particle. Collisions between particles, for example, are no longer necessarily viewed as some variant of billiards-ball behavior, but instead, as the interference of wave functions giving rise to effects much like interference phenomena in optics.
Schrodinger himself offered one of the first interpretations: he argued that the electron is not a particle, but it is a matter wave as the ocean wave is a water wave. According to this interpretation, the particle idea is wrong or only an approximation. All quantum objects, not just electrons, are little waves -- and all nature is a great wave phenomenon. The matter-wave interpretation was rejected by the Gottingen group led by the German physicist Max Born (1882-1970). They knew that one could count individual particles with a Geiger counter or could see their tracks in a Wilson cloud chamber. The corpuscular nature of the electron -- the fact that it behaved like a true particle -- was not a convention. But what, then, were the waves?
It was Max Born himself who answered that perplexing and crucial question. His interpretation marks the end of determinism in physics and the birth of the God-who-plays-dice physics. It occurred in June, 1926, three months after Schrodinger's paper, and it profoundly disturbed the physics community. Born interpreted the de Broglie/Schrodinger wave function as specifying the probability of finding an electron at some point in space. What Born said was that the square of the wave amplitude at any point in space gives the probability of finding an electron there. For example, in region of space where the wave amplitude is large, the probability of finding an electron there is also large. Similarly, where the wave function is small, the probability of finding the electron is also small. The electron is always a true particle and its Schrodinger wave function only specifies the probability for finding it at some point in space. Born held that the waves are not material, as Schrodinger wrongly supposed; they were waves of probability, similar to actuarial statistic giving the probable location of individual electrons. The description of the motion of quantum particles is inherently statistical; it is impossible to track them precisely. The best that a physicist can do is to establish the probable motion of a particle.
Quantities that characterize the dynamical aspects of the state of motion of a particle are called dynamical variables. Some examples of these are position, linear momentum, angular momentum, and energy. In classical physics, these quantities are ordinary numbers or vectors; in quantum mechanics, they are represented by abstract objects called operators, which act upon and transform state functions according to a definite and known set of rules. But the algebraic rule that govern their behavior are totally different from those governing the behavior of ordinary numbers. Specifically, that algebra is noncommutative, which is to say that the result obtained depends upon the order in which the operations are carried out. Let us use the conventional symbols x, p, L and E to denote these dynamical variables of position, linear momentum, angular momentum, and energy, respectfully. The operational nature of these quantities is defined by the effect that each has when it acts upon, and thereby transforms, a certain state function ψ. Thus, for example, p operating upon ψ produces some new state function φ. The essential nature of this transformation is that operating with a dynamical variable is a symbolic representation of an ideal measurement of the variable in question. That is, the transformation of the state function, which results when an operator acts, is the result of the necessary disturbance caused by such a measurement. The noncommutativity of two operators is more or less a statement about the mutual interference between the measurement of the different quantities. For example, consider a particle in a given state ψ. Measurement first of its position x then of its momentum p gives a different result than if these measurements are carried out in reverse order. The difference of order of the position and momentum operators give different results because these measurements interfere with each other.
The variable quantity that describes de Broglie waves is called wave functions, denoted by the symbol ψ (the Greek letter psi). The value of the wave function associated with a moving body at the particular point x, y, z in space at the time t is related to the likelihood of finding the body there at that time. The wave function has no direct physical meaning. There is a simple reason why the wave function cannot be detected by experiment. The probability of experimentally finding the body described by the wave function at point x, y, z at time t is proportional to the value of ψ2 there at t. The probability P that something be somewhere at a given time can have any value between two limits: 0, which is the certainty of its absence, and 1, which is the certainty of its presence. (A probability of 0.2, for example, signifies a 20 per cent chance of finding the body there at that time) Since the amplitude of the wave function may be negative as well as positive, and probability cannot be negative, the wave function cannot represent the probability that something be somewhere at a given time. But this is not true of ψ2, the square of the wave function. For this reason and other reasons ψ2 is called probability density. And the probability of finding the body described by the wave function ψ at the point x, y, z at time t is proportional to the ψ2 there at time t. A large value of ψ2 means the strong probability of the presence of the body, while a small value of ψ2 means the slight probability of it presence. As long as ψ2 is not actually 0 somewhere, there is a definite chance, however small, of detecting it there. This is the interpretation of the wave function that was first made by Max Born in 1926.
[1] Louis De Broglie, Matter and Light: The New Physics
(New York: W. W. Norton & Company, Inc., 1939.
Reprinted by Dover Publications), pp. 46-47.