cphil_qm1
Quantum Mechanics
by Ray Shelton
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.
The wave theory of light uses three concepts: wavelength λ, frequency f, and the velocity of propagation c. The wavelength and velocity can be determined experimentally. The frequency f associated with light of wavelength λ can be found by the wave relation
The study of light is concerned with three basic processes of emission, propagation, and absorption of light. In the emission of light, light energy is produced from some other form of energy and in absorption the light energy is converted to some other form of energy. In fact, light is only detected when it falls on some absorbing object such as the human eye, a strip of film, or a photoelectric cell. In these three cases the light energy is converted into a electric nerve pulse, or causes a chemical reaction, or the release of photoelectrons. In the emission and absorption of light, measurements have indicated that light energy is emitted or absorbed in quantum units, called photons, rather than continuously. The quantum or photon theory is used to explain the emission and absorption of light, and the wave theory to explain the propagation of light. These theories complement each other, since they apply to different processes and describe different aspects of light. Both theories are needed to explain the nature of light. Some have proposed that the photons maintain their identity along the path of the light propagation from the emitter to the absorber. But there is no experimental evidence for this hypothesis. This hypothesis leads to the paradox: how can light be both particles and waves at the same time? A wave is a continuously varying quantity and a particle is discontinuous or discrete quantity. The electric field intensity can be decreased gradually from some finite value to zero. A photon, on the other hand, cannot be subdivided; a fraction of photon is no photon. Einstein proposed that photons have an energy
where h is Planck’s constant (6.625 × 10-34 joule-sec) and f is the frequency of the light. And if photons have energy, then they may be considered to have mass associated with, or equivalent to, that energy. Let designate this “associated mass” of the photon m. Note that the photon does not have a rest mass m0 since there is no such thing as a photon at rest. The photon either moves at the speed of light, or it has been absorbed and no longer exists. Using Einstein mass-energy equivalence relationship,
(mc2)2 = (mv)2c2 + (m0c2)2 or E2 = p2c2 + E02,
where E = mc2 is the total energy of the particle and p = mv is the momentum of the particle, and E0 = m0c2 is the rest energy of the particle. Since the rest mass of the photon is zero, let E0 = 0, then
E2 = p2c2 or E = pc.
That is, the momentum of the photon is
since by equation [1] c = λ f, and by equation [2] E = hf.
And since E = mc2 and E = hf, then the “associated mass” of the photon is
These relationships for energy ( equation [2]) and momentum ( equation [3]) of a photon is experimentally supported by the Compton effect. In these experiments, when a high-frequency radiation, such as a beam of X-rays, strikes an electron, the electron may receive the whole energy of the photon, as in the photoelectric effect, or the electron may recoil with only a fraction of this energy. In the latter event, the photon is scattered with reduced energy and frequency, and the whole process corresponding to a collision between particles having unequal masses. In the collision, the photon may be regarded as having lost an amount of energy that is the same as the kinetic energy K gained by the electron, though actually separate photons are involved. If the initial photon has the frequency f0 associated with it, the scattered photon has the lower frequency f; that is, where:
Loss in photon energy = gain in electron kinetic energy,
Momentum, unlike energy, is a vector quantity, involving direction as well as magnitude, and in the collision momentum must be conserved in each of two perpendicular directions. The directions we chose here are that of the original photon and one perpendicular to it in the plane containing the scattered electron and the scattered photon. The initial photon momentum is hf0/c, the scattered photon momentum is hf/c, and the initial and final electron momenta are respectively 0 and p. In the original photon direction (the x-direction),
Initial momentum = final momentum,
hf0/c + 0 = (hf/c)cos φ + p cos θ, [6]
and perpendicular to this direction (in the y-direction),
0 = (hf/c) sin φ – p sin θ, [7]
where angle φ is between the direction of the initial and scattered photons, and angle θ is between the directions of the initial photon and the recoil electron. From equations [6] and [7], a formula (equation [8]) can be derived, relating the wavelength difference between initial and scattered photons with angle φ between their directions, both of which are easily measurable quantities by means of Geiger counter.
Δλ = λ′ – λ = (h/m0c)(1 – cos φ), [8]
where λ′ = c/f and λ = c/f0.
This equation [8] was derived by Arthur H. Compton (1892-1962) in the early 1920s, and the phenomenon it describes, which he was the first to observe in 1923, is known as the Compton effect. It constitutes very strong evidence in support of the quantum theory of radiation. Equation [8] gives the change in wavelength expected for a photon that is scattered through the angle φ by a particle of rest mass m0; it is independent of the wavelength λ of the incident photon. The quantity h/m0c is called the Compton wavelength of the scattering particle, which for an electron is 0.024 Å (2.4 × 10-12 m). From equation [8] the greatest wavelength change that can occur will take place for φ = 180°, when the wavelength change will be twice the Compton wavelength h/m0c.
In September, 1923, Louis Victor de Broglie (1892-1987) presented two papers that became his doctoral dissertation in which he proposed that with the motion of any electron or material particle there is associated a system of plane waves, such that the velocity of electron is equal to the group-velocity of those waves. He later wrote
“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:
E = hf ( equation [2]) and
p = h/λ ( equation [3]).
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
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
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
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,
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,
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.
De Broglie showed that his hypothesis provided a very simple interpretation of Bohr’s quantum condition for stationary states of the hydrogen atom. That condition was that the angular momentum of the electron orbiting about the nucleus of the hydrogen atom should be a whole-number multiple of h/2π, that is, nh/2π. But if r denotes the radius of the orbit and p the linear momentum of the electron, the angular momentum L is rp. So the condition is
L = rp = nh/2π,
or
2πrp = nh,
and using equation [9b], λ = h/p, we get
2πr(h/λ) = nh, or
2πr = nλ,
where λ is the wavelength of the de Broglie wave associated with the electron and 2πr is circumference of the circular orbit of radius r. That is, this equation means simply that the circumference of the orbit of the electron must be a whole-number of multiple of the wavelength of the de Broglie wave. Thus an electron can circle a nucleus indefinitely without radiating energy provided that its orbit contains an integral number of de Broglie wavelengths. If the circumference of the electron orbit contained a fractional number of wavelengths, then destructive interference would take place as the wave traveled around the orbit and the wave would die out rapidly. This interpretation of the Bohr’s quantum condition is decisive in understanding of the atom. It combines both the particle and wave properties of the electron into a single statement, since the electron wavelength is computed from the orbital speed required to balance the electrostatic attraction of the nucleus. While these antithetical properties can never be observed simultaneously, they are inseparable in nature.
There is no analog in Newtonian mechanics to the wave phenomena called diffraction. In 1927, C. Davisson and L. Germer in the United States and G. P. Thomson in England independently confirmed de Broglie’s hypothesis of wave associated with a particle by experimentally demonstrating that electrons show diffraction when they are scattered from crystals whose atoms are spaced sufficiently.
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 probability”, 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.
One of de Broglie’s thesis examiners knew Einstein and passed the thesis to him, who in turned recommended it to another colleague, Erwin Schrodinger (1887-1961). Few people paid attention to the thesis, but Schrodinger changed all that. He developed De Broglie’s ideas mathematically and published in March, 1926, a single equation, now called Schrodinger’s Equation, purporting to explain all aspects of the behavior of electrons in terms of De Broglie’s waves. This was the beginning of wave mechanics.
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
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
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:
Using the Hamilton’s energy equation [14] above, the linear momentum of the particle p may be expressed in terms of energy (E – V):
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]
As an example of wave mechanics, consider a particle restrained to move in a box. If it cannot get out, this means that the walls represent an impenetrable potential barrier. Since there is no particle outside the box, there will be no wave either; that is, ψ has to be zero outside the box, and so the wave is confined to the box as is the particle. Since the particle is in the box, then the wave will have an amplitude somewhere in the box that is not zero. The lowest energy that the particle can have corresponds to the longest wavelength, which in this case is limited by the length L of the box. If we assume that in the box the potential energy V = 0, then its total energy is its kinetic energy ½mv2.
But since λ = h/mv,
then v = h/mλ,
and the kinetic energy K is given by
K = ½mv2 = ½m(h/mλ)2 = m(h2/2m2λ2) = h2/2mλ2.
The longest wavelength that the wave can have in the box (remember that the wave must fall to zero at the walls) is 2L, where L is the width of the box. This gives us as the minimum energy of the particle
Kmin = h2/8mL2.
The energy of the particle can be raised by shortening the wavelength. So the next shorter wavelength is 2L/2, and the next shorter is 2L/3, and so on. Thus in general the wavelength is 2L/n, where n is any integer 1, 2, 3, … and the kinetic energy of the particle becomes
K = n2h2/8mL2.
In classical mechanics the particle can have any energy, including zero. But when its behavior is described using the wave equation, it is found that the energy cannot be zero, and it is quantized by the number n. Wave mechanics leads naturally to quantization — a feature that before had to be postulated. And it leads to answers different from those supplied by classical mechanics. Practically, h is so small and m is so large for an observable particle that the minimum kinetic energy is too small to be observed. But it might be observable if L is very small.
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.
As another example of wave mechanics, consider a particle in a shallow potential well. As we have seen, according to equation [20], the linear momentum p of a particle depends upon the energy of the particle:
p = √[2m(E – V)].
Since the wavelength λ of the particle is given by λ = h/p, then the particle’s wavelength is
λ = h/√[2m(E – V)].
This equation shows that the wavelength depends upon the particle’s location to the degree that the system’s potential energy V depends upon the particle’s location. According to classical mechanics, unless the particle has energy sufficient to overcome the potential hole in which it finds itself, it must remain in the well. It is meaningless to speak of a particle being at a location for which the total energy E is less than the potential energy V — that would imply a negative kinetic energy or an imaginary wavelength. In wave mechanics an important difference is found. In wave mechanics a particle may be found outside the limits defined by classical mechanics. The solution of Schrodinger’s equation shows that the wave function decreases exponentially in regions for V > E. In classical physics a particle whose kinetic energy is less than the height of the potential well would never get through to the other side simply because of energy conservation. But in wave mechanics this is a possibility. The wave function decays exponentially when the potential energy V exceeds the total energy E. Thus the wave function of a particle approaching the wall from the left is sinusoidal to the left of the wall, exponential through the wall, and sinusoidal, but of much smaller amplitude, to the right of the wall. The particle, or more properly, the wave can penetrate, or tunnel through, the classically insurmountable barrier. This effect is observed in the behavior of certain semiconducting devices (tunnel diodes) and in the emission of alpha particles from heavy, unstable nuclei.
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.
But exactly what is the significance of these particle wave functions? Should we think of an electron as a localized ball of stuff, or as some extended wave? And if it is a wave, what is waving? After all, there is no such thing as an aether light wave; a light wave is a handy paraphrase for time and space varying electric and magnetic fields. Are they matter waves? And what, then, is a matter wave?
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 dynamical variables used in classical mechanics, such as position and momentum, do not have definite values in quantum mechanics. Instead they operate on a quantity called a “wave function” into which is encoded probabilistic information about position, momenta, energies, etc. Thus in quantum mechanics the motion of particles is not deterministic, as in classical mechanics, but probabilistic. The wave function for a particular system is found by solving the Schrodinger equation.
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.
Consider the interference effects that arise in the passage of waves through two parallel slits. When either one of the two slits is closed, the pattern is the typical single-slit diffraction pattern: a broad, central maximun flanked by weaker, secondary maxima. When both slits are open, the pattern is an inference fine structure within a diffraction envelope. The pattern is not merely two single-slit diffraction patterns displayed; it is the result of the interference of the waves coming from both of the slits; the interference between the waves traveling through both of the slits is responsible for the rapid variations in intensity. Interference of waves follow from the principle of supperposition. The resultant wave at any point is the sum of the individual wave contribution from the sources – in this case the waves coming from each slit. Thus at points on the screen where the individual waves have the same phase, the resultant wave has a relatively large amplitude. At points where the individual waves are out of phase, the resultant wave has a relatively small amplitude. Since the intensity is proportional to the square of the amplitude, the interference pattern contains alternating maxima and minima of intensity. In short, in a case in which waves can take two or more routes from a source to an observation point, the problem is solved by first superposing the wave functions from the two separate routes to find the resultant wave function and then second by squaring to find the resultant intensity. Applying this to de Broglie wave functions, if ψ1 and ψ2 represent the probability distribution for passage through slits 1 and 2 separately, then
(ψ1 + ψ2)2,
not ψ12 + ψ22,
gives the probability of observing a particle on the screen. If, then, a single electron is directed toward a pair of slits, we cannot say which of the two slits it will pass through; we must speak in the language of waves and say that it passes through both slits. The electron is behaving in a nonlocalized way, like a wave. There is no logical contradiction involved in these complementary explanations. If the experiment is arranged so to observed the particle-like aspects of an electron behavior, it exhibits those aspects instead of the wave characteristics. Conversely, if the experiment is arranged to observe its wave-like aspects, then the electron will exhibit those aspects instead of the particle characters. The electron is not just a particle nor just a wave. It is both. In the double slit experiment electron exhibits its wave characteristics. Quantum mechanically, when both slits 1 and 2 are open, the probability amplitude is the sum of the individual probability amplitudes:
ψ12 = ψ1 + ψ2,
and the probability ψ122 is thus given by
ψ122 = (ψ1 + ψ2)2 = ψ12 + 2ψ1ψ2 + ψ22.
The middle term, the cross product term, is responsible for the interference. It can be either positive or negative depending upon the relative signs of ψ1 and ψ2. When ψ1 and ψ2 have the same sign, constructive interference takes place, and the probability is enchanced over its classical value. When ψ1 and ψ2 have opposite signs, destructive interference takes place, and the probability is reduced from its classical value. The fact that probability amplitudes add together, just as do the amplitude of water waves or light waves, is the essential reason for the occurrence of interference. This principle of superposition, as it is called, lies at the heart of quantum mechanical description of the physical world. We conclude that when an event can occur in two or more alternative ways, the resulting probability amplitude is the supperposition of the probability amplitude for each separate event. Labeling these alternative ways by 1, 2, 3, and so on, the net probability amplitude is
ψ123… = ψ1 + ψ2 + ψ3 + …
and for the probability itself
(ψ123… )2 = (ψ1 + ψ2 + ψ3 + …)2.
The cross terms obtained when this last expression is expanded manifest themselves observationally in the phenomena of interference. (The mathematical treatment here of wave functions has been simplified. Actually, the state function ψ at a given point is not a ordinary real number, but is what is called a complex number, a + ib, where i is the imaginary unit = √-1, or i2 = -1. This means that two real number are required to specify ψ at a given point. This makes the rule for calculating probabilities accordingly more complicated than has been given here.)