Reconning with uncertainty
Familiar quotations
“Quantum mechanics is very impressive. But an inner voice tells me that this is not yet the right track. The theory yields much, but it hardly brings us closer to the Old One’s secrets. I, in any case, am convinced that He does not play with dice.” Albert Einstein1
“I think I can safely say that nobody understands quantum mechanics.” Richard Feynman2
This is an imagined 3 dialogue in three parts between an inquisitive and persistent student and some of the noteworthy minds responsible for developing quantum mechanics.
Part I, Forerunners
Experimentalist (J. J. Thompson, 1856-1940): I’ve been able to show that the filament of a cathode ray tube emits particles having a very small mass and a negative charge. Let’s call these corpuscles (later electrons). They were ejected from the atoms of the heated cathode filament and were electrically guided through a vacuum to a distant fluorescent screen where they could be detected as a light flash.
Student Questioner: That is interesting; I wonder what else may be hidden within the atoms of your heated metal filament, or, for that matter, within other forms of matter.
Experimentalist (Ernest Rutherford, 1871-1937): Apparently, atoms contain mostly empty space. Fellows in my research laboratory discovered this when they aimed positive alpha particles at a gold foil. Most of the time, the alpha particles passed through the foil unimpeded. You may know that alpha particles have a much larger mass than an electron but a much smaller mass than a gold atom.
Student Questioner: You stated that most of the time they penetrate through the gold foil unimpeded …?
Experimentalist (Rutherford): As it happens, very occasionally an alpha particle strikes something massive within the foil. In this case, the alpha particle is sharply deflected. Because it travels through unimpeded most of the time, it seems to me that nearly all the mass of the gold atoms must be concentrated within a small portion of that atom’s total volume, and that this concentration of mass must have a positive charge.
Student Questioner: So, you think that the concentration of mass within the atom has a positive charge?
Experimentalist (Rutherford): Apparently so, because on the rare occasion when the alpha particle, which has a positive charge, closely approaches, or strikes something in a gold atom, it is repelled. We know that like charges repel. Let’s name the concentrated mass the “nucleus” of the atom.
Student Questioner: Since the foil seems to have no intrinsic net charge, I would assume that electrons must balance the positive charge of the gold atoms’ nuclei. But if most of the space surrounding the nucleus is “empty,” what are your thoughts on how electrons occupy this space?
Experimentalist (Rutherford): I’m not sure, but perhaps they circle the nucleus like planets around our sun.
Theoretician (Neils Bohr, 1885-1962): That can’t be right, though. James Maxwell’s equations clearly show that an orbiting charge would produce electromagnetic radiation. If electrons were racing around the nucleus, they would quickly radiate away all their energy. But atoms have been around for a long time, and these electrons (corpuscles) have not lost their energy.
Neils Bohr
Student Questioner: OK, but how do electrons avoid falling into the nucleus if they’re not circling it?
Theoretician (Neils Bohr): J.J. Thompson has already determined the mass/charge ratio for electrons. He showed that electrons have a very small mass but a substantial negative charge. Having a negative charge, they should be attracted to the positively charged nucleus, and they require kinetic energy to avoid being pulled in. If they possess kinetic energy, they must have orbital momentum. I wonder if this dilemma is explained using some of the new ideas proposed by Max Planck regarding energy quantization.
Student Questioner: Well, I’m curious about your thoughts on that.
Theoretician (Neils Bohr): Admittedly, we don’t know much about the state of electrons within atoms, but we do know that when they are pulled away from their atom, they may be detected on J. J. Thompson’s detection screen having a coating of florescent material. Let’s begin by trying to describe these electrons as having energy and orbital angular momentum when in the atom.
Student Questioner: OK, I’m interested in the direction this is leading.
Theoretician (Neils Bohr): I’ll make a trial calculation. Assume that only some values of the electron’s energy are allowed ― like what Dr. Max Planck has proposed for the energy of electromagnetic radiation. Let’s say that in the atom, an electron’s energy, and its orbital angular momentum, 𝓛n, are quantized. I’ll propose that the angular momentum of electrons obey the relationship:
𝓛n=nh/2π . (1)
Student Questioner: I can understand the n because this allows you to assign different discrete values for the angular momentum, somewhat like what Dr. Planck proposed for the quantized energy of electromagnetic radiation. The other terms I don’t understand.
Theoretician (Neils Bohr): I borrowed the constant “h ” from Dr. Planck. Planck’s constant for quantization, h , has units of angular momentum. The 2π relates to a circular orbit. Dr. Planck and Dr. de Broglie will have more to say on this later. For now, I’ll try to put these ideas into an equation for an electron’s energy when it is trapped in the atom.
Student Questioner: OK then. I’ll wait … maybe you can construct something that makes sense to me.
Theoretician (Neils Bohr): I’ll calculate an electron’s energy for the simplest atom, hydrogen, and try to explain data of the light energy emitted from an energized hydrogen gas sealed within a glass tube. A Swiss mathematician, Johann Balmer, has published a description of this data. He put these experimental results into the form of a mathematical series.
First, I’ll assume that the single electron of a hydrogen atom may lose or gain energy, in other words change its orbital “radius” and angular momentum. I’ll calculate this gain or loss of energy as absorbed or emitted light. To do so, I’ll need a value for the orbital “radius.” Using quantization of angular momentum in equation (1) together with classical equations relating an electrons orbital radius to its centripetal force and acceleration, I calculate this “radius” for the electron’s orbit as
r = n2h2 𝜖0/(𝜋mee2) = n2x 0.53x 10-8 cm , (2)
or roughly half an angstrom for the lowest value of n = 1 . This is much larger than Rutherford’s estimate of the size of the concentrated nuclear mass. I propose this as an estimate of a hydrogen atom’s size.
If the electron remains in orbit, then its kinetic and electrical potential energies would have a fixed relationship. So, I determined the kinetic energy , K.E. , using the centripetal force on the electron: F = me· acceleration = meeV2/r = (1/4𝜋𝜖0)e2/r2 ; it follows that K.E. =½meeV2 = (1/8𝜋𝜖0)e2/r . From electrostatics, the electron’s potential energy in a hydrogen atom, P.E. = (1/4𝜋𝜖0)e2/r . So, the total energy of the electron of a hydrogen atom, using the orbital radius of equation (2), is
K.E. + P.E. = T.E. = -mee4/(8𝜖02 h2n2 ) = -13.6 eV/n2 . (3)
This equation quantitatively gives the energies of radiated light from an excited hydrogen gas as empirically described by Dr. Balmer. As the electron’s orbital energy changes from a higher to a lower value (from n to n’ ), the emitted light has energies E and frequencies 𝜈 ( E = h𝜈) that are predicted by equation (3).
Student Questioner: Well, that is very impressive, Dr. Bohr. You may be onto something. I have a question, though. You previously mentioned the difficulty of pinpointing an electron’s position within the atom. If, as you stated, an electron orbiting its nucleus would lose energy through radiation, then the electron that you are proposing would soon lose all its energy. Can this be so? Are you trying to have it both ways with your description?
Theoretician (Neils Bohr): You are right. Someone will need to come up with a more general description.
Another Theoretician (Erwin Schrödinger, 1887-1961): I intend to address this problem immediately.
Erwin Schrödinger
Student Questioner: May we back-track first, though? Dr. Bohr proposed that angular momentum is allowed only certain values. This directed him towards equation (3,) which allows only certain values for the electron’s total energy. This appeared to explain the emission spectrum of an excited hydrogen gas, in which the radiated light has only certain energies seen as discrete frequencies.
Are there other examples which support this notion that energy levels allow only certain discrete values?
Theoretician (Max Planck, 1858-1947): Certainly, I first made the case for energy quantization of electromagnetic radiation. I did this to explain the spectral radiance of light emitted from the tiny orifice of a heated solid. This phenomenon is termed “black body” radiation. I found that a “black body’s” radiation behaves as though the released light energy were allowed to have only discrete values, E𝜈 , which are related to the emitted light’s frequency, 𝜈:
I suggested that the black body’s radiation of energy cannot be continuous, but rather that it must exist in discrete energy amounts, or energy quanta of the size h𝜈. I was able to calculate a value for h . Others have named this “Planck’s constant” in my honor. Dr. Bohr used the parameter, h , in his model for the angular momentum of an electron in a hydrogen atom. Though I originated this idea, I was among those who were reluctant to entirely accept energy quantization. By nature, I am known to be conservative in my views.
Even though my formula was empirical, I was pleased that it accurately described the light spectrum of a heated body and that it was also useful for Dr. Bohr’s description of the hydrogen atom’s emitted light and the orbiting energy of its electrons. These early examples were an indication that energy may be quantized, or at least an indication that this could be the case under some conditions.
Max Plank
Student Questioner: Are there other examples?
Theoretician (Albert Einstein, 1879-1955): Well, I found a way to explain how light is capable of ejecting electrons from metal surfaces. This is known as the photoelectric effect, and it helped me win the Nobel Prize. I postulated that light behaves as if it exists in packets or drops of energy. In 1926, Gilbert Lewis named these “photons.” I followed Planck’s lead and gave each photon of light an amount of energy represented in equation (4). I was able to explain previously published results, which had shown that electrical current may be generated by light. This turned out to be a useful experimental and theoretical finding.
Albert Einstein
Student Questioner: Well, that’s two examples. Are there others?
Experimentalist/theorist (A.H. Compton, 1879-1962): This was a hot topic in the early 1900s. I was also awarded the Nobel Prize for explaining my experiments of X-rays striking electrons in the carbon atoms of a compressed block of graphite. I needed the photon concept of Planck and Einstein with E𝜈 = h𝜈 and Einstein’s special relativity. (Because X-rays have such high energy, the impacted electrons are deflected with relativistic velocities.) My analysis included the energy of the incident photon, h𝜈, the energy and direction of the scattered photon, h𝜈‘ , and that of the scattered electron. Using momentum and energy conservation, the shift in frequency, Δ𝜈, of the scattered photon could be predicted. So, X-rays appear to have quantized energy too.
Student Questioner: OK, it seems convincing to me that different types of electromagnetic radiation are quantized with energies h𝜈. I am also impressed that the h in this expression is the same as was used to quantize an electron’s angular momentum in the Bohr model for hydrogen. Something seems fundamental about the constant h . What is not satisfying, though, is this notion of the position of the electron. It remains fuzzy to me. Has there been any progress in generalizing these ideas?
Theoretician (Louis de Broglie, 1892-1987): I proposed that the momentum, p , of any particle, including an electron, could be assigned a wavelength, p = h/𝜆 . This definition for momentum was inspired by Maxwell’s radiation pressure, Einstein’s expression for the momentum of light, p = E/C, and Planck’s energy quantization: E = h𝜈. Particles with smaller momentum have larger wavelengths. In an atom, a tiny electron, which I suggest is a quasi or pseudo particle, is best described by a standing wave having wavelength 𝜆. The precise position of the electron seems to have little meaning. Defining an electron’s momentum in terms of its wavelength obviates the position (or radiation) problem and gives a rationale for Bohr’s concept of discrete angular momentum states. Angular momentum quantization (equation 1) is equivalent to requiring that the wavelength of an electron fits neatly as a standing wave in a circular pattern around its nucleus. This explains the 2π in equation (1). The idea won me the Nobel Prize when I was quite young.
Part II, Formalism
Student Questioner: Thank you for returning to the conversation. I have more questions, as you may have predicted.
I can accept that something as small as an electron, which one would never be able to see, may have a wavelength. And I accept that an electron’s standing wave should fit neatly within an atom, as in the Bohr model for hydrogen. But I have heard of other concepts in quantum mechanics, such as uncertainty. Einstein was quoted as saying that it seemed to him as God playing with dice, and he apparently didn’t much like the uncertainty of quantum mechanics. Could someone explain the concept of uncertainty?
Theoretician (Erwin Schrödinger): I have made progress on this subject. However, my description, though very general, is highly mathematical and abstract. The usual intuitive interpretations of classical physics don’t seem to apply in this realm of small distance and small momentum (large wavelengths). I developed an equation that relies only on probabilities; it inherently includes uncertainties, explicitly avoiding exact values. I’ll call this new approach wave mechanics. Probabilities are embedded within a wave function, 𝛹. The central equation for the wave function is
𝑖ħ𝜕𝛹(x,t)/𝜕t = -(ħ2/2m)𝜕2𝛹(x,t)/𝜕x2 + V(x,t)𝛹(x,t) = E𝛹(x,t) . (5)
(The symbol ħ = h/2𝜋 . )
Student Questioner: Your equation looks exotic to me with imaginary numbers and this symbol 𝛹(x,t). What is the meaning of a probability wave function for describing a particle like, for example, an electron?
Theoretician (Schrödinger): Here is the rub: we may not know. We do know that wave equation (5) will predict experimental results, and, for this reason, it has proven to be very useful.
Student Questioner: Can you at least give some help in beginning to think about the wave function?
Theoretician (Schrödinger): Perhaps this would be a useful suggestion: Assume that the potential V (x,t) in the wave equation (5) depends only on position. This will be true for many problems you’ll face. In this case, the wave function, 𝛹(x,t), may be described as a product of a time part and a space part as
𝛹(x,t) = 𝜓(x) · 𝜙(t) . (6)
After substituting equation (6) into equation (5), equation (5) becomes two equations, one for position (7) and one for time (8).
-(ħ2/2m)d2𝜓(x )/dx2 + V(x )𝜓(x) = E𝜓(x) . (7)
E represents the system’s total energy including its kinetic and potential energies. Equation (7) is an ordinary second-order differential equation describing the position portion of the wave function, 𝜓(x). A solution will depend on the potential energy function, V(x), the value of E , the factor, ħ2/2m, and any required boundary conditions for the wave function.
The time-dependent portion of the wave function must satisfy the equation
d𝜙(t)/dt = -𝑖E 𝜙(t)/ħ . (8)
Equation (8) has general solutions that one might encounter in courses on differential equations, and it has a deceptively simple one that will become useful (equation 9). Time is assumed to have no boundaries and the total energy, E, is required to be a real number, as we shall see.
𝜙(t) = exp(-𝑖E t/ħ). (9)
Student Questioner: So far, I’m not seeing how this helps.
Theoretician (Schrödinger): I return to the earlier theoretical work of Dr. Bohr, in which he calculated the transition between an electron’s energy levels for the hydrogen atom. For bound states, the total electron energy is less than zero. For the spatially bound (or confined states) states of Bohr’s electron in a hydrogen atom, energy must be quantized. I’ll generalize and assume energy to be quantized if a particle is confined within a spatially bound state.
So, for equation (9) to be compatible with Bohr’s hydrogen atom model, and Plank’s energy quantization, there would need to be multiple possible solutions, 𝜙n(t ), each with a different quantized energy level, En .
𝜙n(t) = exp(-𝑖Ent/ħ) . (10)
There are other constraints that should be considered. For example, in equation (10), the function 𝜙n(t) must not go to infinity as time goes to infinity. So, as previously mentioned, energy, En, should be a real (not an imaginary) number.
A complete solution for 𝛹(x,t) in equation (6) for the spatially bound electron states must allow different possible energy levels, En, and different possible time-dependent solutions 𝜙n(t). One must solve the time-independent portion, 𝜓n(x), for each allowed energy level, En. First normalize the wave function to be sure of the particle’s existence somewhere in space and then combine this with the time-dependent part, 𝜙n(t). This sounds simple, but it may become mathematically tedious.
Student Questioner: OK this seems a bit much for me right now. Does it help with understanding the meaning of the probability wave?
Theoretician (Schrödinger): Try thinking of the time-dependent portion of the particle wave, 𝜙n(t), as like the time-dependence of an electromagnetic wave. Just as each photon of light has a pulsating nature with frequency, 𝜈 = E/h , each particle, depending on its energy level, would have a pulsating nature with a quantized frequency 𝜈n = En /h =𝜔n /2𝜋 . To illustrate this, substitute the expression for quantized energy and frequency into equation (10).
As for the time-independent (spatial) part, consider replacing momentum with its wavelength equivalent. In developing equation (5), I used a differential expression for momentum, incorporating the ideas of de Broglie for the kinetic energy part, into a Hamiltonian-like expression. As a result, a particle wave will be described by a combination of wavelengths and frequency components.
Student Questioner: It may help a little, Dr. Schrödinger. However, Dr. Feynman’s quotation continues to come to mind. Nevertheless, could someone say more about the time-independent part of the wave function and perhaps elaborate on these discrete energy levels?
Figure from the internet: An illustration of the position, x, and momentum, p, representations of a particle using wave the functions ψ(x) and ϕ(p). The real part of these wavefunctions is illustrated as Re[ψ(x)] for position; for momentum it is Re[ϕ(p)]. The relationships p =h/λ, Δx·Δp = h/2π, and this figure may aid in visualizing the reciprocal relationship between quantum mechanical position and momentum. Compare the upper row where position is better defined and the lower row where momentum is better defined. Also, see the Appendix.
Theoretician (David Hilbert, 1862-1943): This discussion is going to become even more mathematical now because I am primarily a mathematician. I’ll need to define some things first. As you likely know, mathematicians require precise definitions.
Student Questioner: Please don’t be so precise in your definitions that I won’t be able to understand anything you may say. So, what do you need to define?
Theoretician (David Hilbert): I’ll start with eigenvector and eigenfunction. An eigenvector is a generalization of something familiar. The three basis vectors in three-dimensional Cartesian space are orthogonal eigenvectors, êx, êy, and êz .
Generalize this to multi-dimensional space with the eigenvectors ê𝑖 (𝑖 = 0 ⟶ n ), with no limit on the integer value of n. In analogy with the three dimensions of Cartesian space, the eigenvectors of multi-dimensional space are orthogonal vectors of unit magnitude. Using the abbreviations of linear algebra and the bracket notation ⟨ , ⟩ for an inner product, the orthonormality requirement may be expressed as
⟨ê𝑖 ,ê𝑗 ⟩ = ê𝑖 · ê𝑗 = 𝛿𝑖𝑗. (11)
The Kronecker delta, 𝛿𝑖𝑗 , is zero unless 𝑖 = 𝑗, in which case it equals one.
Student Questioner: OK, but how does this relate to wave functions and discrete energy levels?
Theoretician (David Hilbert): I need to expand the eigenvector concept and use the term eigenfunction. I’ll take the time-independent portion of the wave equation, 𝜓(x), and write this as a summation of eigenfunctions, 𝜓n(x):
𝜓(x) = cn 𝜓n (x) . (12)
If an index appears twice in the same term as in equation (12), this implies a summation over all allowed values of that index. Albert Einstein introduced this convention.
There are many examples of the deconstruction of functions into their orthogonal components, which are well known to mathematicians. One is credited to the great French mathematician Joseph Fourier, who used sine and cosine components for his deconstruction of spatially limited functions ― that is, those functions having strict spatial boundaries.
The symbol 𝜓(x) represents the full spatial wave function; the symbol 𝜓n(x) represents one of its components or eigenfunctions. The weights, cn , assigned to each component eigenfunction can be viewed as equivalent to coordinates in an eigenfunction space, like the coordinates of a vector in three-dimensional space. The totalities of these weights are required for a full representation of 𝜓(x) as shown in the summation of equation (12). Just as a position in a three-dimensional space may be described by its geometrical coordinates, a total wave function may be described by the weights (or coordinates, cn) of constituent eigenfunctions in a multidimensional space. As is the case for eigenvectors (equation 11), we require orthonormality of the eigenfunctions.
⟨𝜓*𝑙(x), 𝜓𝑚(x)⟩ = 𝛿𝑙𝑚 (13)
Because a quantum mechanical wave function may have imaginary parts, the proper way to form the inner product shown in equation (13) is using the function’s complex conjugate designated by the asterisk, 𝜓*𝑙 (x). This will be implied in all expressions involving the bracket notation ⟨ , ⟩.
Eventually, as in equation (6), the spatial components of a wave eigenfunction, 𝜓n(x), must be combined with the temporal components, 𝜙n(t) for each allowable energy level, En , to complete the wave function. If there are multiple spatial components for a single energy level n, then the energy eigenvalue, En , is said to be “degenerate.” An example of degeneracy in the hydrogen atom is represented by the several angular momenta, 𝓛 , eigenfunctions, 𝜓𝑛𝓛 (x), or angular momentum states of the electron for each energy level, En , in the hydrogen atom.
Student Questioner: Yes, I suppose I can see how this might help with describing multiple angular momentum states for each energy level of electron orbitals in atoms, but could you help me understand how your formulism could be useful for solving general problems and predicting outcomes?
Theoretician (Hilbert): OK, suppose you want to predict a measurement of an observable parameter. In quantum mechanics, you need an operator for this parameter. For example, the operator, 𝒫 , for momentum, p , is expressed as
𝒫 = -𝑖ħd/dx (14)
The momentum operator may be seen within the kinetic energy portion of Schrödinger’s equation (7), where kinetic energy p2/2m =(- ħ2/2m)d2/dx2. For the position parameter, we designate the generic operator, using the symbol, 𝒳.
Student Questioner: Could you illustrate with an example?
Theoretician (Hilbert): Because of the orthonormality requirement for the eigenfunctions, the solutions may seem simple, but you should probably work this through for yourself.
In general terms, though, the position portion of a particle’s wave function is given by equation (12). Because a particle must exist somewhere, the expectation for its existence (the probability for it having a place somewhere in space) must equal unity. For the spatial portion of the wave function, this requirement is expressed using eigenfunctions and the bracket notation of linear algebra:
⟨c𝑙 𝜓𝑙(x), c𝑛 𝜓𝑛 (x)⟩ =∑n [cn ]2= 1 (15)
The expectation value for the result of a generic operator, 𝒬 is expressed as
⟨𝒬⟩ = ∑𝑛q𝑛[c𝑛 ]2, (16)
where q𝑛 , the eigenvalues, are defined as 𝒬𝜓𝑛⟩ = q 𝑛𝜓𝑛 and the bracket lines [ ] signify absolute values. The eigenvalues are the result of an operator, 𝒬, acting on a single eigenfunction. Equation (16) may be expanded as follows.
⟨𝒬⟩ = ⟨𝜓(x) , 𝒬𝜓(x)⟩ =
⟨c𝑙𝜓𝑙 (x), 𝒬c𝑛𝜓𝑛(x)⟩ = ⟨c𝑙𝜓𝑙(x),q𝑛c𝑛𝜓𝑛 (x)⟩ = ⟨∑𝑙 c𝑙𝜓𝑙(x) , ∑𝑛 q𝑛c𝑛𝜓𝑛 (x)⟩ = ∑𝑛q𝑛[c𝑛]2. For the last two expressions, the summation symbol rather than Einstein’s summation convention is used. Orthogonal wave functions require that only terms where 𝑛 = 𝑙 are counted. Do not confuse the eigenvalues, q𝑛, with the components (or coordinates), c𝑛, of the eigenfunctions.
Student Questioner: So, for the calculation of the most likely, or the expectation value, of a measurement, say of position among the range of all possible positions that could occur, would one square the weights, [c𝑛]2, of each of the constituent eigenfunctions, 𝜓𝑛(x) , multiply this by the result, q𝑛, of the operator for position acting on each eigenfunction, and then take the sum? Would the most likely value of the measurement be near the statistical average of the many measurement outcomes (eigenvalues) that are possible? (The average, then, may not line up precisely with one of the allowed quantum values for position, I suppose.)
Theoretician (Hilbert): Yes. It is as if the full wave function were representative of a statistical “population” of eigenfunctions. An individual eigenfunction is called a determinate state: that is, one eigenfunction among a population of many eigenfunctions. The result of the operator acting on a single determinate state (single eigenfunction) is a single eigenvalue, q𝑛, of the operator. Also, an eigenvalue represents one of many possible outcomes of a measurement. The expectation for each measurement value would be related to the weight that is assigned to that individual outcome. Measurement of any specific one, even the most probable one, is never guaranteed. Herein lies the uncertainty of quantum mechanics.
Student Questioner: This uncertainty remains my sticking point. Could you elaborate?
Theoretician (Hilbert): I remember that your original question was about uncertainty. Though uncertainty is built into the formulations for quantum mechanics championed by both Schrodinger and Werner Heisenberg, their approaches to uncertainty were different. Schrodinger’s uncertainty lay in the wave function, while Heisenberg’s uncertainty lay in the operators for the observables…momentum and position, for example. In quantum mechanics, the ultimate result of the action of an operator on a wave function may be interpreted only as probability, which is the chance that a specific measurement result may occur. This probability could be the chance of a particle being found in each of the possible configuration states, [c𝑛]2. For example, the position operator for each constituent eigenfunction would place the particle at a specific place, but that place represents only one of many that are possible. By the definition of normalization, the summation, ∑𝑛[c𝑛𝜓𝑛]2= ∑𝑛[c𝑛]2 = 1, (equation 15), which is a statement that the sum of all the individual probabilities, must equal 100%.
It’s important to note, however, that the result of two operators acting in succession may depend on which one acts first. Take for example the operator for position (𝒳) and for momentum (-𝑖ħ)d/dx . The full wave function with all its possibilities will be changed by either of these operators acting. So, when the second operator acts, it will not find an untouched wave function. Any measurement will change the profile of the eigenfunctions. For example, if a measurement gives a specific result q𝑛’, then the wave function will have collapsed to the corresponding single eigenfunction state, 𝜓𝑛’ ; so now, c𝑛’ = 1; all the other c𝑛 = 0 , (if 𝑛 ≠ 𝑛’). Everything will have been changed by this measurement.
It becomes even more exotic, though: A particle’s wave function may have more than one representation. There exists a representation using position (the position space representation). But there is also a representation using wavelength (the wavelength space representation). Wavelength space describes the wave functions with one of three compatible parameters: either its wavelength (𝜆), its wave number (k), or its momentum (p) note 4. These three parameters are all compatible with each other, but they are incompatible with the particle’s position. Either momentum or position may be used, and either descriptor has its valid wave functions, eigenfunctions, and a valid space in which these eigenfunctions are allowed an existence.
However, even though a quantum mechanical wave function may be converted from a position space representation to a momentum space representation, the eigenfunctions for position and momentum are never simultaneously applicable (Appendix). One certainty in quantum mechanics is that attempting to “pin down” a particle with a precise position will blur its wavelength (momentum) properties and vice versa. My colleague Werner Heisenberg emphasized this important property. As an example, if the electron in an atom is said to possess a well-defined momentum (wavelength), it has no position. So, one needn’t have worried about an electron radiating away its energy as Neils Bohr may have once pondered for the hydrogen atom. While within the atom, an electron has no unique position — only a well-defined wavelength!
Student Questioner: Is it necessary that the set of eigenfunctions for position and for momentum are never simultaneously available? This seems baffling to me.
Theoretician (Hilbert): Yes, it is true even though somewhat baffling. Let’s return to the basics. In quantum mechanics, just as in classical mechanics, we require mathematical systems to predict measurements involving, for example, position, time, momentum, or energy. In classical mechanics, these are given by calculus and by Newton’s laws of motion, or by Hamilton’s or Lagrange’s methods. In the much smaller, unseen, realm of quantum physics, we must use an operator for each measurement, a probability wave function to describe the particle, and a mathematical space, in which that probability wave function and the eigenfunctions of the particle are allowed. A measurement can be viewed as causing the wave to collapse to one of its eigenfunctions. As it turns out, and intrinsically associated with the uncertainties of quantum mechanics, there is no so-called “Hilbert space” containing sufficient eigenfunctions for the simultaneous measurements of momentum and position. In other words, a particle wave cannot collapse to a single momentum eigenfunction and a single position eigenfunction simultaneously.
Momentum and position operators “do not commute.” If the order of the operations is reversed, there will be a different outcome mathematically and a different measurement experimentally.
Physicists have honored me by naming the space for eigenfunctions as the “Hilbert space.” In quantum physics, particle waves are said to “live” in Hilbert space, allowing for example, a space for momentum or a space for position, but not both. It must have allegiance to one or the other –no dual citizenship is allowed in the quantum mechanical world.
Student Questioner: So, is that all you really need to know to grasp quantum mechanics…?
Part III, Quantum descriptions extend deeper into the unseen.
Theoretician (Richard Feynman, 1918-1988): You should know that quantum mechanics includes more than has been described here … even more mysteries lie within this fascinating topic. Dealing with this subject requires a certain amount of humility, I believe. I once had said that nobody understands quantum mechanics; nevertheless, I received the Nobel Prize for using quantum mechanics, mathematics, and my special diagrams to describe how light, matter, and charged particles interact. In quantum mechanics, each of these forms of energy, light, matter, charged particles, and their interactions may be described using waves or “vibrations in space”.
However, long before my work, others had extended the foundations of quantum mechanics. For example, scientists who preceded me described an electron’s spin angular momentum and proposed an exclusion principle for electrons. I’m not saying that we fully understand it, but electrons belong to a class of particles for which exclusion is an absolute requirement. That is, double occupancy of the same wave function space by two identical electrons is strictly forbidden. Without an exclusion principle, many electrons could crowd into the same atomic orbital and the variety of, and the chemical properties of elements of the periodic table would be lost. Our universe would then be boring, and we would not be here to ask these questions. However, two electrons, distinguishable only by their different spin properties must be allowed to occupy a single wave-function space. This is necessary as two electrons occupying such a space are required to form the chemical bonds between atoms. This property makes the chemistry of our world possible.
[As an aside, other constraints on particles such as the quantum-mechanical coupling of two-photon or electron-positron pairs are thought to persist. These relationships termed entanglements remain even at great distances, as has been recently demonstrated experimentally. This is in complete agreement with the uncertainty of quantum mechanics, but some claim that it confounds Einstein’s prediction that no information can be transmitted faster than the fundamental speed of light. Following the friendly disagreement between Einstein and Bohr at the fifth Solvay Conference in Brussels, quantum mechanics seems to be presently ahead.]
Theoretician (Richard Feynman, 1918-1988): If you’re satisfied by these answers so far, though, you probably have not been paying much attention. Even without a full understanding, you should become familiar with the techniques and terminology of quantum mechanics if you wish to be conversant with physical descriptions of the very tiny, unseen, and mysterious, spaces of our universe. Our comprehension of these phenomena has certainly increased since my contributions were made, and I expect that future progress will continue. And though breakthroughs may come in bursts, more often, I suspect that they will develop slowly over time.
Student Questioner: Do you have any further parting wisdom?
Theoretician (Richard Feynman): Before I conclude and even though it is becoming late, let me elaborate on one important property of electrons previously mentioned ― their spin angular momentum ― let me describe how this has come to our present-day understanding.
Stern and Gerlach were experimental physicists in the early 20th century. Though it was not immediately recognized at the time, their experiments revealed that when within an atom, an electron’s magnetic character, which we attribute to its spin, must be quantized. Observations (pinning down) an electron’s spin restricted its value to one of only two possible outcomes. The theorists, Goudsmit and Uhlenbeck later provided us with an explanation of Stern and Gerlach’s experiments.
We associate an electron’s spin with its magnetic moment and, by association, with the magnetic moment of its host atom. It may be incorrect to describe the magnetic properties of electrons and atoms using only a gross analogy with magnets as magnets represent a collection of many individual magnetic elements. Nevertheless, these elementary properties in atoms give rise to the behavior of magnets that we observe in everyday life. One would not be surprised to find that in the confined quantum mechanical space of an atom, an electron’s magnetic moment would obey exotic quantum rules; one such rule is that electrons are allowed only one of two possible measured values for their spin.
This property of quantized electron alignment was eventually recognized from the previously mentioned results of Stern and Gerlach’s experiments using silver atoms. Each silver atom contains a single unpaired electron. When propelled through an uneven magnetic field and examined, the silver atoms behaved as though only one of two possible magnetic alignments were found; hence only one of two possible observed “spin states” was apparently allowed: either spin “up” or spin “down.”
Wolfgang Pauli stated the rule that when two electrons are observed in a state of the same wave function, then they must possess another “hidden” internal quantum-level descriptor to differentiate them. For two electrons in a double occupancy wave function, one must have a designation spin + ½, and the other must have a designation spin – ½. This rule is an example of the Pauli Exclusion Principle. Fermions, the class of elementary particles for which this logic applies, were named after Enrico Fermi, another famous physicist of the last century. Electron paring influences the magnetic properties of atoms that we observe and use every day.
Student questioner: Is this the final description for sub-atomic particles?
Theoretician (Richard Feynman): Certainly not, but rarely in life, or in science, are descriptions final. Though, sometimes they may seem simpler, like when Newton, Maxwell, or Einstein took seemingly unrelated experimental observations and consolidated them into simpler, more compact, and more “elegant” theories. Physicists constantly search for these.
Usually, though, the deeper one looks, the more complexity one finds. This has perhaps been the case for the quantum description of matter so far.
Exotic theories have been proposed. Though, every scientist realizes that understanding must be backed by experimental testing of one’s hypotheses. Any other approach would be “rubbish, as James Clerk Maxwell once lectured to his students in Scotland. Utilizing increasingly powerful machines and astronomical observations, physicists keep trying to examine matter more deeply in hopes of arriving at simple and more compact descriptions.
(Optional) Appendix I: Heisenberg uncertainty
Heisenberg’s uncertainty principle is fundamental to physics and goes beyond specific examples. Many of the teaching examples in quantum mechanics are of bound states of matter, like electrons within an atom, or of particles constrained within a box. In these, quantization appears. However, even for a particle that is not trapped in a bound state and for which energy and momentum may not be quantized, the uncertainty principle applies. One way of observing this mathematically is using a Gaussian wave function and the mathematics of Fourier transforms. A hypothetical Gaussian waveform packet for a particle represented in position space may be expressed as
𝜓(x ) = exp (-𝛽x2) 𝛽>0 . (A1)
For the Fourier transformation of this waveform packet into a Hilbert space based on the parameter k or, equivalently, momentum, p , or wavelength, 𝜆, one may use:
k = 2𝜋/𝜆 = p/ħ . (A2)
This transformation using the parameter k yields
[𝜓(k)]transform = exp(-k2/4𝛽)/√(2𝛽) . (A3)
(The transformation is identified by the brackets around the wave function. You may check this result by using the definition of a Fourier transform and a table of integrals.) Focus now on the relative width of the wave packets for position and momentum. Putting equations (A1) and (A3) into the standard form of a Gaussian wave packet requires from (A1) that
𝛽 = 1/(4𝜎x2) . (A4)
But from (A3) we must also have that
𝛽 = 𝜎k2. (A5)
Equating these expressions for 𝛽 [(A4) and (A5)] reveals that the range (width) for a wave packet in k space, 𝜎k ,multiplied by the range (width) of the wave packet in position space, 𝜎x , must equal ½:
𝜎x ⋅ 𝜎k = ½ . (A6)
Taking 𝜎x as an estimate of the minimum range for x , Δx ; 𝜎k as the minimum range for k, Δk = Δp/ħ, equation (A2), yields the Heisenberg’s uncertainty relationship for the minimum of the product ΔxΔp
ΔxΔp ≥ ½ħ . (A7)
A good relationship to remember in quantum physics is that p = h/𝜆 = hk/2𝜋 = kħ.
(Optional) Appendix II: Follow-up questions
Question: Does quantum mechanics require all wave functions to be represented by quantized parameters?
Answer: Apparently, no, as a “free” particle in infinite space may not have quantized energy or momentum. This is related to the fact that one expects the wavelength of a “free” particle to exist within limitless space. However, particles that are confined within fixed limits have quantized energy and momentum.
Question: Look carefully at the Schrodinger equation previously shown:
𝑖ħ𝜕𝛹(x,t)/𝜕t = – (ħ2/2m)𝜕2𝛹(x,t)/𝜕x2 + V(x)𝛹(x,t) = E𝛹(x,t) .
- What part of this equation represents the kinetic energy of the particle?
Answer: – (ħ2/2m)𝜕2𝛹(x,t)/𝜕x2
- What part represents the potential energy?
Answer: V(x)𝛹(x,t)
- What parts represent total energy?
Answer: E𝛹(x,t)
- If we assume that the potential energy is negative and the kinetic energy is positive, will the total energy be positive or negative … free or bound?
Answer: This depends on the sign of the total energy
- In the case of the hydrogen atom, what is the sign of the total energy in Schrodinger’s equation?
Answer: Negative
Question: Must photons originate from a bound state of matter? Consider, for example, an excited hydrogen atom, a “black body” cavity, a harmonic oscillator of charged particles, or a linear accelerator of charged particles.
Answer: After the origin of the universe in the so called “big bang,” photons obey E = h𝜈. In atoms, black body radiation, or a harmonic oscillator, light emerges from a bound state and is quantized in packets of h𝜈. The cosmic background radiation emitted at the origin of the universe comes in packets too, so one may conclude that the emergence of the universe may too have been from a confined space. Additional student research may be needed for clarity on this.
Question: Describe the relationship between Einstein, Hilbert, Bohr, Heisenberg, and Planck… how their lives were intertwined, and how they were affected by the Nazi takeover of Germany.
Answer: This will require additional biographical research.
References and Source Material
Footnotes
1Albert Einstein, letter to M. Born, December 4, 1926; The Born –Einstein Letters. P. 90
2Feynman Lectures: chapter 6, “Probability and Uncertainty — the Quantum Mechanical View of Nature,” p. 129
3Written for the purpose of physics pedagogy, not claiming historical accuracy.
4This equivalence is consistent with the definitions of de Broglie.
Source Material
Erica W. Carlson, Understanding the Quantum World, The Teaching Company (2019)
David Halliday and Robert Resnick, “Physics, Parts I and II”, John Wiley & Sons, Inc. (1960)
David J, Griffiths, “Introduction to Quantum Mechanics”, 2nd Edition, Pearson Education (2005)
Eugen Merzbacher, “Quantum Mechanics”, 2nd Edition, John Wiley & Sons, Inc. (1970)
Serge Lang, Linear Algebra, Addison-Wesley (1967)
Abraham Pais, “‘Subtle is the Lord …’ The Science and the Life of Albert Einstein” Oxford University Press (1982)
Richard Feynman, “‘Surely, you’re Joking, Mr. Feynman!’ Adventures of a curious Character” Bantam Books (1986)
J.L. Heilbron, “The Dilemmas of an Upright Man, Max Plank as Spokesman for German Science” University of California Press (1986)
Editors, the Charles River Press, “Niels Bohr, The Life and Legacy of the Influential Atomic Scientist”
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