Introduction

 

Classical vs. Quantum Theory

      Originally, chemistry was viewed through only a classical perspective. Theories were created about macroscopic objects, and were confirmed by experiments on macroscopic objects. But when these theories were applied to very small, quantum, objects, theories fell through.

     One such theory was Rayleigh-Jean’s Law. It was known that objects emitted blackbody radiation. This meant that a blackbody, which was a non-reflective object, would emit electromagnetic radiation. Rayleigh-Jean’s Law predicted the intensity of the radiation to the temperature of the blackbody, and the wavelength of the emitted light. At large wavelengths this theory matched the experimental outcomes very closely. When tests were performed on very small wavelengths, the Law predicted that the intensity would raise exponentially, which is not demonstrated by experiments. This was known as the ultraviolet catastrophe (see picture below).

 

Blackbody radiation experimental results vs. Rayleigh-Jeans predictions

     This image demonstrates the experimental results for the intensity of blackbody radiation at low wavelengths as peaking, then tapering off. This is juxtaposed to the predictions made by Rayleigh-Jeans’ law, which demonstrated exponential growth. From this scientist began to consider quantum objects to demonstrate different properties.

 

Particles vs. Waves

     In what is known as the “stone-age” of chemistry, pre-quantum, it was believed that things were either particles or waves, but not both. With this theory, an electron, which has mass, would be a particle, and light was a wave. Einstein’s photoelectric effect was one of the first theories to break this conception, by demonstrating that light had particle like characteristics.

     In Einstein’s photoelectric effect light was shone onto a surface. If the light had a low frequency, nothing happened, and the light was simply absorbed into the surface. When the light had a high frequency, the surface emitted electrons, where the kinetic energy of the electrons had a linear dependence to the frequency of light shone on the object. This theory tied together the wave frequency of light, a wave-like characteristic, to energy, a particle-like characteristic. Einstein demonstrated that light transferred energy in particle-like bundles, which were named photons.

Several experiments further disproved the separation between field and particle characteristics. Young’s Double Slit experiment for example, demonstrated that light had wave-like characteristics as well. In this experiment light were shone at a surface with two small slits. If the light acted only like particles, it should have passed through the small slits, and continued to travel straight to the detector. The detector would then only record two spots where light hit it. In actuality it was found that light was detected in a diffraction pattern, similar to waves.

Diffraction: evidence of light's wave-like behavior

 

     Therefore it was concluded that light had both wave-like and particle-like characteristics. But if light could be wave-like and particle like, could matter have the same properties?

 

Electrons as Waves and Particles

     Once it was established that light can act as both a wave and a particle, the question was shifted to electrons. Could electrons also be both wave-like and particle-like? It was already established that electrons could be particle-like, they had mass! An experiment was performed, where electrons were shot at an aluminum film. When x-rays were shot at the film, the waves defracted and created a circular diffraction pattern. With particles, we would expect the particles to either bounce off o the aluminum, or to shoot right through. But when electrons were shot a diffraction pattern similar to those of x-rays were found.

 

Diffraction patterns of electrons, proof of electrons' wave-like properties

 

     Bohr was the first to suggest the wave-like characteristics of electrons. He hypothesized that angular momentum could only have discrete values, and that electrons could move only in discrete orbitals. This theory was supported by the atomic emission spectra, which demonstrated that atoms only absorbed discrete frequencies of light, and they were unique to each element. Bohr’s model, which was only made for a hydrogen atom, was further examined by Planck, who made a model that generalized for all atoms. But what could cause electron’s energy to be quantized?     

     Like a guitar string, energy is quantized when there are standing waves. From here it was hypothesized that electrons moved in a wave pattern around the nucleus of the atom. The waves around the nucleus were standing waves, which meant ‘n’ number of full wavelengths had to fit in the circumference of the circle.

 

Electron travelling as waves around the nucleus

 

de Broglie connected the wavelengths of the electron to the velocity at which they are traveling, through his formula

 

This connected waves and particle characteristics into one formula!

 

Heisenberg’s Uncertainty Principle

     Once it was established that electrons act in both particle and a wave fashion, several things became even less clear! de Broglie demonstrated that p= h/wavelength. But this wave is spread uniformly around the circle, therefore this formula does not tell us anything about the angular position of the electron. Warner Heisenberg proposed that there is a limit to what we can know about a function at one time. He identified that certain qualities, such as energy and time, or length and momentum could not be measured definitively at the same time. From this arose the Heisenberg uncertainty principle formula;

(ΔA)(ΔB) ≥ 1/2|[A, B]|

where ΔA and ΔB are the uncertainty in A and B respectively. Let’s say that A is length and B is time, and these cannot be known completely certain at the same time. Then if we know everything about length, in order for this inequality to be true, ΔB will be very large, meaning we will be very uncertain about the time. By Heisenberg’s theory, uncertainty became an intrinsic part of systems. Even at absolute zero, where theoretically nothing is moving, there must be a little movement and a little momentum, in order for there to be uncertainty about both at the same time.

 

Basic Understanding

 

Operators & Commutators  
     To better understand Heisenberg's uncertainty principle, we must first consider operators - what they are, and their function in the Heisenberg equation.      

     An operator takes a function and performs a mathematical process on it. They are often expressed with a capital letter with a hat.

 

The operator M expresses a multiplication operation between the numbers in parentheses.  

 

The nature of the mathematical process of a particular operator must be defined for each case. Operators are especially useful for expressing wave functions. Operators provide an elegant way to express the otherwise complicated mess generated by integrating and differentiating wave functions. 

Using operators simplify the expression of Schrodinger's equation

     

  The commutator is result of operations between operators. It is an operator itself, and can be expressed as such:

 

  The commutator should not be confused or associated with the commutative property. The commutative property is a basic property encountered in math.

 

Commutative property of addition: a + b = b + a

Commutative property of multiplication: a*b = b*a

 
The commutative property can be extended from its theoretical basis to real life cases. For a given day, I can choose to watch Youtube videos and also amuse myself with Internet memes. Whether I do one first or the other, the outcome is the same - a fun but ultimately unproductive day.

     It’s important to note that whether for theoretical or real cases, order doesn’t matter - the resulting outcome will be the same. However, the commutative property does not apply to the product of operators. The product of operators is not commutative, that is, A B =/= B A, unless the commutator goes to zero, in which case, the operators are said to commute.

     Operators are said to commute if their commutator vanishes, or in mathematical terms, when it goes to 0. It can be mathematically expressed as such:

 

 

     The Heisenberg uncertainty principle, (ΔA)(ΔB) ≥ 1/2|[A, B]|, then represents a commutator. The following figure shows that the uncertainty equation, with position and momentum operators, is true by finding the commutator [x, p].  

 

First, we define our position and momentum operators as such:

x^ψ(x) = xψ(x)

p^{^}ψ(x) = -iℏd/dx ψ(x)

We can calculate the commutator of [x^,p^] = [x, -iℏd/dx] for some function f(x):

[x, -iℏd/dx] f(x) = -iℏ(x d/dx - d/dx x) f(x) 

distribute the function f(x):

    =-iℏ(x d f(x)/dx - d/dx (x(f(x))

take the derivative, remembering to take the chain rule for the second part:

=-iℏ(x df/dx - f(x) - x df/dx)

the terms x df/dx and -x df/dx cancel:

=iℏ f(x)

 

We find that [x, p] = iℏ, and does not equal 0. 

 

The operators of the Heisenberg uncertainty principle do not commute, and this fact has rather important implications. Two operators that do not commute cannot be measured simultaneously, and so for the HUP, it means that it is impossible to know exactly the values of position and momentum; this uncertainty is an intrinsic part of the system. For the momentum and position operators to commute would imply that both momentum and position could be precisely measured at the same time, which for small particles, is an event that cannot exist in nature.

 

The Heisenberg uncertainty principle is quantitatively given as: 

(Δx)(Δpx) ≥ ℏ/2

where Δx and Δp represent the probabilities of position and momentum respectively. 

 

 

Applications for Understanding

 

     The Heisenberg Uncertainty Principle, sometimes referred to as the Indeterminacy Principle to avoid any misconceptions, is often difficult for one to intuitively wrestle with as a physical manifestation in our macroscopic world, relative to quantum entities. However, one of the most easily observable results of the Uncertainty Principle was observed as early as 1828 by John Herschel. A science renaissance man of sorts, he documented for his article concerning light for the Encyclopedia Metropolitana. The phenomenon was only fully formalized from a theoretical approach in 1835 by the Astronomer Royal Sir George Biddell Airy in his “On the Diffraction of an Object-glass with Circular Aperture.” This phenomenon has been dubbed The Airy Pattern or the Airy Disk.

 

The Airy pattern phenomenon

 

     This characteristic distribution of light is due to the wave attributes and the associated wave interference of the massless, particle-like photon and the distribution of the photons create the distinctive pattern. The pattern is the consequence of shining a beam of light through an aperture and the resultant photon striking the canvass, or wall, creating the pattern.

     At a glance, the Airy Pattern seems only to further demonstrate the wave-particle duality concept that accompanied the rise of quantum mechanics. However, we can vary the diameter of the aperture, which, in turn, varies the diameter of the Airy Disc. This is analogous to what you already know about the Heisenberg Uncertainty Principle, which states that there is an inexorable limit to the precision with we can both measure and know the momentum and position of a quantum entity. Think of the size of the aperture as a measure of the uncertainty of the attribute position, and the diameter of the Airy Disk as a measure of the uncertainty of the attribute momentum. Keep in mind that the diameter of the aperture is inversely proportional to the diameter of the Airy Disk Pattern, that is, the smaller the aperture, the bigger the pattern, and vice versa.

     For example, assume that the aperture is very small; we have a very good idea of the position of the quantum entity and have measured it with a small uncertainty (small aperture). The resulting Airy Disk is rather large; we have a poor idea of the momentum of the quantum entity because we know the position so well and consequently have measured it with a large uncertainty (large diameter). Additionally, the inverse holds no matter how non-intuitive it seems: a large aperture (large uncertainty in position) produces a smaller Airy Disk (small uncertainty in momentum because of large uncertainty in position).

 

This image is of a quantum harmonic oscillator with an initial Gaussian distribution and it illustrates the probabilities densities of two variables and is equally as analogous to the Uncertainty Principle as the Airy Pattern analogy. Think of the red curve as representing position and the blue curve as momentum, with the area under the curves, or the integral of the curves, representing the certainty with which we know each value. Notice the inverse relationship of the two areas (or certainty) and how one must decrease if the other increases. Together, they represent the fundamental limit to how precisely we can measure and know each value at any given point in time.

 

Time-Energy Uncertainty Principle

     The applications of the Heisenberg Uncertainty Principle do not end at the mathematical model of the intrinsic limit with which to determine the precision of two of a particle's attributes. Other uncertainty relationships have been discovered as well, but one of most erroneously cited, on a general scale, is that of the time-energy Uncertainty Principle, which is as follows:

ΔtΔE≥ℏ2

        The mathematical relation seems sound at a glance, but this relationship can only be derived under special circumstances. A sensible interpretation of the expression quickly reveals the absurdity of the relationship in its application outside of these special circumstances. For example, some have said that the uncertainty of time can somehow be related to the uncertainty of energy and that this has applications regarding the ability to circumvent the law of energy conservation. You have to realize that the probability distribution for the attributes being related in the expression (ΔA and ΔB) do not change with time, but with each other. In this instance, energy serves as the "time" scale because it changes with reference to physical observables, or things we can measure in experiments. Energy serves as the "time" scale because the term Δt has been determined from physical observables, such as momentum for example. These physical observables are measured over time. Since the uncertainty relationship operates independently of our conventional perception of time, we could not possibly have time as a variable (Δt≠ΔA)! However, notice that we can relate time to observables, which is exactly what you do when you run an experiment in the laboratory-you take measurements of the observables with respect to time such as position vs. time.

     Now let's take a look at how the Time-Energy Uncertainty Principle has been derived and under what circumstances it is useful. The important thing here is to realize that time alone is not enough, but we need observables with which to relate time to the Uncertainty Principle. We must think about time, Δt, as being derived from and related to physical, measurable observables or the relationship simply does not make sense, practically, mathematically, or theoretically. Remember that a relationship between time and observables make sense because we often measure them with respect to time.

Starting with the general form which can be used to describe any state, with values of energy Ek, and eigenvectors |k⟩:

|ψ⟩=kck|k⟩≡|ψ⟩=ncnψn(x)

With t0 representing the initial state, we have the state as a function of t:

U(t,t0)|ψ⟩=kcke-iℏEk(t-t0)|Ek⟩≡U(t,t0)|ψ⟩=ncnψn(x)e-iEntℏ

Now we replace the t variable with something that makes sense on a quantum mechanical level, an observable, A, which will serve to characterize the change in the system in time. We will denote the uncertainty (H), or standard deviation, of A in the already established initial state |ψ⟩by ΔA(the uncertainty of energy will similarly be denoted as ΔE). Utilizing the uncertainty relationship in the initial state we have:

ΔAΔE≥12|⟨[A,H]⟩|

Relating our expression to the commutator for the expectation values through time:

12|⟨[A,H]⟩|= ℏ2ddt⟨A⟩

Substituting for the commutated expression:

ΔAΔE≥ℏ2ddt⟨A⟩

In order to use A to characterize the time scale Δ for a significant change in the system, we can relate the rate of change of the average value of A to the initial uncertainty in A:

Δt=ΔA|ddt⟨A⟩|

Having defined Δt as such, we end up with Time-Energy Uncertainty Principle:

ΔtΔE≥ℏ2

We can perform a thought experiment to test the reasonability of our new uncertainty relationship. Imagine that the initial state is stationary, where the energy eigenvector is 0, giving us ΔE=0. Recall the inverse relationship between the two uncertainty parameters. If ΔE=0, then Δt ∞. If we look back at how Δtis defined, Δtwill go to infinity as|ddt⟨A⟩|becomes very large, meaning the observable will not change. This makes sense because there is no energy in the state.

|ψ⟩=kck|k⟩

There are two interpretations of the Time-Energy Uncertainty Principle and how they relate the state vector, |ψ⟩, and the operator-observable, |k⟩: The Schrödinger Picture and the Heisenberg Picture. In the Heisenberg Picture, the vector that represents the state and is fixed it time, |ψ(t)⟩=|ψ(t0)⟩, while the operator-observables evolve in time(|k(t)⟩≠|k(t0⟩). The opposite is true in the Schrödinger Picture, which dictates that the state vector does change through time, |ψ(t)⟩≠|ψ(t0⟩, while the operator-observables remain fixed ((|k(t0)⟩= |k(t)⟩).

 

Concept Questions

 

1. What is the significance of operators not commuting in the Heisenberg uncertainty principle? 

 

2. How does the Airy diffraction phenomenon support the HUP?

 

(answers given after references)

 

 

References

 

Campion, Gillis, Oxtoby. Principles of Modern Chemistry (6th ed) Thomson Learning, Inc. 2008.

 

Sension, Roseanne. "Chem 260/261 - Lecture 5: The Birth of Quantum Mechanics." Chem 260/261 Lecture. University of Michigan, Ann Arbor. 14 Sept. 2012. Lecture.

 

Sension, Roseanne. "Chem 260/261 - Lecture 6: Quantum Laws." Chem 260/261 Lecture. University of Michigan, Ann Arbor. 17 Sept. 2012. Lecture.

 

http://depts.washington.edu/chemcrs/bulkdisk/chem455A_aut10/notes_Lecture%2012.pdf

 

http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Commuting_Operators

 

http://www-inst.eecs.berkeley.edu/~cs191/fa07/lectures/lecture14_fa07.pdf

 

http://www.rhythmodynamics.com/Gabriel_LaFreniere/sa_Huygens_files/airy00.jpg

 

http://ocw.usu.edu/physics/classical-mechanics/pdf_lectures/14.pdf

 

http://upload.wikimedia.org/wikipedia/en/6/68/Position_and_momentum_of_a_Gaussian_initial_state_for_a_QHO%2C_wide.gif

 

Answers

 

1. The fact that the operators of the HUP do not commute signifies that the variables cannot be measured simultaneously, giving rise to intrinsic uncertainty of a system, which the HUP defines. 

 

2. The Airy diffraction shows the relationship of position and momentum given by the HUP. The diameter of the aperture equates to the uncertainty of position and the diameter of the Airy disk equates to the uncertainty of momentum. When the diameter of the aperture is small, there is small uncertainty in position, and the diameter of the Airy disk, which represents the uncertainty of momentum, becomes greater, and vice versa.