216                        The Bible According to Einstein

Now the second way to understand quantum mechanics is through the quantum wave equation. And although it is not obvious, this approach shall be entirely equivalent to the path-integral approach.
     And let it be postulated that there be a wave function for an entity or object. And let it be further postulated that the wave function provides the probability of finding the entity at a particular position. Thus the wave function sums and incorporates all information of the quantum paths. Thus the wave function directly gives the position and momentum probabilities. And so the wave function, like the path integral, incorporates uncertainty. And to obtain the wave function, one must solve the quantum wave equation.
     Now a wave function for an object shall be like unto a cloud of varying intensity. And the brighter, whiter regions shall indicate locations where the object is most likely to be seen. And places where the object is less likely to be found shall correspond to fainter, less dense regions of the cloud. And if the wave function vanishes somewhere, then cloudy mist shall not be there – no chance exists to find the object there. So this cloud shall not be an ordinary one, but a cloud of probability – intensity shall correspond to probability.
     And although uncertainty shall prevail, quantum mechanics shall specify the uncertainty exactly. And if the wave function, which by definition is a solution to the quantum wave equation, is known, then the probability of finding an object at a particular place at a particular time shall be exactly known.

Chapter VII: Quantum Waves

Now to understand the wave-function formulation one has to understand a wave. A wave is just an undulating motion such as the rise and fall of ocean water.
     Now a wave shall have a highest point, which shall be called the crest. And a wave shall have a lowest point, which shall be called the trough. And let it be known to thee and all around thee that, unlike particles, waves shall interfere, and this shall be their most important property. And for example, when two water waves merge, their crests can coincide and the height of the wave can be increased – this shall be constructive interference because the displacement is enhanced. On the other hand, when two waves merge, the crest of one can coincide with the trough of the other and the waves can cancel – this shall be destructive interference because the wave motion is diminished.
     Now wave functions shall function like unto water waves. And the most likely position of an object shall coincide with a point of maximum displacement such as at a crest or trough. And there shall be no chance of finding an object at a position where the wave is not displaced – such a point is sea level for ocean water waves.

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     And since wave functions act as water waves, wave functions too can interfere. Thus quantum waves shall undergo constructive and destructive interference. An object shall more likely be at constructive-interference points. And an object shall less likely be near destructive-interference points.

Chapter VIII: Wave-Particle Complementarity

And the size of a wave unit shall be called its wave length – the distance between two crests. Now the length of a quantum wave shall have a special meaning – it shall determine when a wave function functions like a wave and is quantum mechanical and when a wave function functions like a particle and is classical. When the quantum wave length is long, the wave function shall be like a wave. And when the quantum wave length is short, the wave function shall be like a particle and not be like a wave.
     Now when the quantum wave length is short, the spatial probability distribution shall be localized – the object must be near a tiny spot. And so the position of the object shall be known with greater certitude; there shall be little uncertainty, and classical mechanics shall be a good approximation. And the wave function shall behave like a classical object such as a macroscopic ball or block. And in this case when the wave length is very small, it shall be hard to see the crests and troughs. And it shall be difficult for such a wave to interfere constructively or destructively with another wave because its crests and troughs are so closely spaced. Such a wave function shall be like the wave around a fast-moving small torpedo – the entity shall behave more like a bullet than a wave. And if such a wave so strikes a wall, the destruction shall be localized in time and space. The impact shall be like unto the impact of a particle against a wall.
     But when the quantum wave length is long, the position shall be delocalized – the object shall be located anywhere within a certain region. And so it shall be difficult to determine precisely where the object is. And constructive and destructive interference shall often naturally occur. And the wave function shall indeed act as a wave. Now when a long wave strikes a wall, it slaps against the wall slowly and the impact shall not be like a bullet or torpedo – this shall be the way a quantum wave collides. And using classical mechanics for such a long quantum wave would be a bad approximation.
     And the contrasting behavior of short and long waves shall be called particle-wave complementarity.
     And when shall an object have a long quantum wave? It shall have a long wave whenever it has low momentum. And when shall an object have a short quantum wave? It shall have a short wave whenever it has high momentum.

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     And since the momentum of an object is proportional to its mass, a heavy body shall always have a short quantum wave and thus behave as if it be a particle. Hence macroscopic bodies, such as blocks and balls and living bodies, never shall behave like waves. Only lightweight entities, such as atoms, nuclei, and subatomic particles, can have long waves. And when such lightweight entities have low momentum, they shall indeed have lengthy quantum waves and be wave-like. Thus, for example, shall the wave functions of low-energy electrons interfere. And when such electrons interfere, they shall create constructive and destructive interference regions – this shall be an electron interference pattern.¹⁹⁶ And since the electrons in the atoms usually have low momentum, such electrons shall behave like waves. And the collection of waves of atomic electrons shall be called the electronic cloud. Now when an electron has high momentum, it shall have a short wave and be like a particle. And high-energy electrons, when they strike a target, shall produce localized, bullet-like destruction.

(And a voice spake, saying
"Beware of multi-MeV electrons and beta radiation.")

Now often there shall be only discrete solutions to the quantum wave equation. And it is like unto the string of a guitar. And when the string is plucked, various vibrational motions are then possible. And if there be only one crest, then the crest appears in the middle of the string. And this is one solution. And if there be two crests, then a still point appears in the middle of the string and the crests appear on either side. And this is a second solution. And there can be vibrations with three crests. And in this case, one crest appears in the middle of the string and the other two emerge on either side of it. And between the middle crest and each side-crest is a still point. Thus there are two still points. And this is a third solution. And in general there can be solutions with any number of crests. And excluding the ends of the string, which are tied down and motionless, the number of still points on the string is one less than the number of the crests. Now for each type of vibration, there is a note. Thus there is a discrete set of sounds. And a guitarist with only these selected notes is unable to play tunes – to create other notes he uses his finger to shorten the string by pressing it against the neck of the guitar. But in the quantum world, there is no such way to shorten waves.

(And a voice spake, saying
"Ye cannot put thy finger on a quantum wave.")

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     Now a single vibrational motion of the guitar string shall in quantum mechanics correspond to what is called a state. A state shall by definition be a particular solution to the quantum wave equation. And a particle shall be in a particular state when it has a particular wave function. And the different notes of the guitar shall correspond to different energies. So each state shall have a unique energy – an energy level shall be the name for this. And since there are only distinct possibilities for states, there shall be only distinct possibilities for energies. And this is why the name quantum mechanics shall be used, for "quantized" means "discrete possibilities."
     And for example, molecules shall have energy levels. And they shall be called molecular energy levels. And atoms shall have energy levels. And they shall be called atomic energy levels. And nuclei shall have energy levels. And they shall be called nuclear energy levels. And so on.
     And among all states shall be one state with lowest energy. And it shall be the ground state.¹⁹⁷ The existence of such a state shall be essential for the stability of matter, for classically, electrons in an atom can lose energy by continually emitting electromagnetic radiation. In such a hypothetical classical world, electrons would spiral into nuclei and atoms would collapse. And once atoms did collapse, solids would fall in upon themselves. Then macroscopic objects such as planets, living bodies, balls and blocks, would so collapse. But in Nature’s quantum world, electrons in atoms shall not spiral into nuclei – instead they shall radiate until they reach the quantum state of lowest energy. And once in this ground state, there shall they remain. And in this manner shall the stability of matter be maintained.

Chapter X: Many-Body Quantum Mechanics

Now when many objects and/or particles are present, there shall be a quantum wave equation for each particle or object. But only one wave function shall there be. And it shall solve all the wave equations simultaneously. And it shall be called the multiparticle wave function. And a multiparticle state shall be a particular solution of the quantum wave equations.
     Now in Nature’s world, there shall be two kinds of particles according to their spin: fermions and bosons.¹⁹⁸ Fermions shall obey the Pauli exclusion principle: It says that two fermions of the same type cannot be in the same state. And for example, all the electrons in an atom cannot "sit" in the ground state. Now the Pauli exclusion principle can be understood as the vanishing of the multiparticle wave function when two identical fermions are "put" in the same place. And why is this so? It is so because a zero wave function means zero probability. Thus it shall be impossible for two identical fermions to occupy the same position or same state.

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     Now identical bosons shall behave the opposite of fermions. Not only shall they try to occupy the same position or same state, they shall do so maximally: The interchange of two selfsame bosons shall not change the state. And the situation shall be similar to that of identical red marbles: After putting all the marbles in a bag and shaking it, one shall no longer know which is which. Bosons are like unto red marbles; the multiparticle wave function for this system is like unto the bag. The indistinguishability of selfsame bosons shall make them behave in the most democratic way. And for example, if two neutral pions’ positions are interchanged, the world shall not be changed.¹⁹⁹
     Now imagine ye a room of seats. Quantum states shall be like unto seats,²⁰⁰ and fermions shall be like unto human beings: Once a person occupies a seat, another cannot sit in that same seat. And so a second person entering the room must search for a different spot, one where no one sits. And if there enters still a third, that person must find a space where the other two are not. And so by analogy, selfsame fermions may not be in the same place. But bosons shall be different; they shall be like unto ghosts: When the "cloudy image" of a ghost occupies a seat, another ghost can sit in that same seat. And if the two sit on top of one another, their cloudy images shall simply make a whiter cloud. And furthermore, it is possible for still a third ghost to sit down where the other two are sitting. Such ghosts do not get in each others’ way in searching for a space. And so by analogy, bosons in the quantum world may be in the same place.

Chapter XI: The Wave Equations

And quantum mechanics shall rule all. It shall manifest itself for objects that are small. Quantum mechanics shall govern atoms, nuclei, and subatomic particles like photons, electrons, neutrinos, quarks and gluons. Now subatomic particles shall obey special quantum wave equations called relativistic wave equations. And larger microscopic objects shall satisfy the ordinary quantum wave equation, the so-called Schrödinger equation.

Chapter XII: Quantum Angular Momentum

And quantization shall apply not only to energy but to other quantum observables. And quantum observables shall be quantities that one can observe and measure, such as position, energy and momentum.

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     And the amount of revolving of an object shall be its angular momentum. And since angular momentum is an observable, it shall be quantized. And the possible amounts of angular momentum shall be limitless but yet discrete. And these amounts shall be multiples of a fundamental unit, which is Planck’s constant. Thus there shall be states with angular momenta of zero, one, two, three, . . . multiples of this fundamental unit.

Chapter XIII: Summary