# Quantum Mechanics - Module 3

# Introduction

Many students are nervous of quantum mechanics, particularly the mathematics used, or think that it is irrelevant to chemistry and biology. In this module we hope you will discover that quantum mechanics is not much more difficult than other topics, that it does apply to the real world and, most importantly, that it is crucial to a fundamental understanding of physics, chemistry, and biology.

If the following picture is your mental image of an atom, with electrons looping around the nucleus, you are about 70 years out of date.

The modern world of quantum mechanics gives us the probability plots (shown below) for where an electron is most likely to be found in the various atomic orbitals of a hydrogen atom (the nucleus is at the center of each plot).

You have seen these plots for years already, but no one has probably ever explained why they should be this way electrons behave. It has just been something that you had to accept as Truth from the wise physicists. The aim of this unit is to reduce the element of faith in your understanding of atomic structure and increase the element of experimental understanding.

Understanding of some of the material in this Module can be assisted by taking advantage of a number of interactive Java applets available on the internet. These exercises are identified by the logo on the left, and will naturally require internet access. There will also be an opportunity at the Residential School to access this material through UNE’s network. Links to all exercises – and additional links – can be found at http://www.une.edu.au/chemistry/CHEM201_QM/CHEM_201.html.

Quantum chemistry applies quantum mechanics to problems in chemistry. The influence of quantum chemistry is felt in all branches of chemistry:

- Physical chemists use quantum mechanics to calculate (with the aid of statistical mechanics) thermodynamic properties of gases (e.g. entropy, heat capacity); to interpret molecular spectra, thereby allowing experimental determination of molecular properties (e.g. bond lengths and bond angles, dipole moments, barriers to internal rotation, energy differences between conformational isomers); to calculate molecular properties theoretically; to calculate properties of transition states in chemical reactions, thereby allowing estimation of rate constants; to understand intermolecular forces; and to deal with bonding in solids.
- Organic chemists use quantum mechanics to estimate the relative stabilities of molecules, to calculate properties of reaction intermediates, to investigate the mechanisms of chemical reactions, to predict aromaticity of compounds, and to analyze NMR spectra.
- Analytical chemists use spectroscopic methods extensively. The frequencies and intensities of lines in a spectrum can be properly understood and interpreted only through use of quantum mechanics.
- Inorganic chemists use ligand field theory, an approximate quantum mechanical method, to predict and explain the properties of transition-metal complex ions.

Although the large size of biologically important molecules make quantum-mechanical calculations on them extremely difficult, biological chemists are beginning to benefit from quantum mechanical studies of conformations of biological molecules, enzyme-substrate binding, and solvation of biological molecules.

This module will be divided into three topics: 3A – Pre-Quantum Mechanics, 3B – Introduction to Quantum Mechanics, and 3C – Model Systems and the Hydrogen Atom. These are derived from the chapter headings in the book ‘Physical Chemistry’ by David Ball, on which this module was originally based. The module will bring us to the gate of Quantum Chemistry, but will not truly take us through, because it is the quantum mechanics of things much more complicated than hydrogen atoms that chemists are mostly interested in. To a good approximation, however, the whole vast menagerie of molecules can be understood by putting together the basic concepts we will derive for the hydrogen atom.

Besides being useful, quantum mechanics can be an exhilarating mental exercise, as it is concerned with the behaviour of entities that are very very different from the macroscopic things we are used to dealing with. Along with these notes you will receive a copy of Chapter 5: The Naked Atom, from “The God Particle” (L. Lederman, Houghton Mifflin, Boston, 1992). On p. 143 Lederman writes:

“But be forewarned! The microworld is counterintuitive: point masses, point charges and point spins are experimentally consistent properties of particles in the atomic world, but they are not quantities we can see around us in the normal macroscopic world. If we are to survive together as friends through this chapter, we have to recognize hangups derived from our narrow experience as macro-creatures. So forget about normal; expect shock, disbelief. Niels Bohr, one of the founders, said that anyone who isn’t shocked by quantum theory doesn’t understand it. Richard Feynman asserted that no one understands quantum theory.”

# Topic 3A

# Pre-Quantum Mechanics

Towards the end of the nineteenth century many physicists felt that all the principles of physics had been discovered; little remained but to clear up a few minor problems and improve experimental methods to ‘investigate the next decimal place’. Those discoveries constitute what is now called classical physics. The early 20th century saw the introduction of the theories of relativity and quantum mechanics. Quantum mechanics was developed by several people over a period of several decades, and it is an extension of classical physics to subatomic, atomic and molecular sizes and distances. Relativity theory and quantum mechanics constitute what is now known as modern physics. Although relativity has had little impact in chemistry (why do you suppose this is the case?) quantum mechanics has played a very important role, so that an introductory course in quantum mechanics and its applications to chemistry - quantum chemistry- is an integral part of any chemistry course.

## Laws of motion – classical mechanics

You might find it quite strange to see material on classical mechanics in a course on quantum theory, especially of quantum theory as relevant to chemistry. But it really is illuminating to revise and discuss several concepts in classical mechanics before embarking on a description of quantum mechanics.

Classical mechanics describes the motion of objects. To see how it does this, we will look at equations describing the constancy of energy and the response of particles to external forces.

### Total energy

The total energy of a particle is the sum of its kinetic energy, K (the energy arising from its motion), and its potential energy, V (the energy arising from its position):

E = K + V

but recall that
*K = *½* m v ^{2}* (m mass; v velocity) hence:

*E* = ½ *m v ^{2} + V*

We can express this in terms of the linear momentum, p = mv:

^{p}^{2}/_{2m}* + V*

Using the fact that v = ^{dx}/_{dt}, we can rearrange this equation into a differential equation for x as a function of t:

*E* = ½*m( ^{dx}/_{dt})^{2} + V*

2*(E - V) = m( ^{dx}/_{dt})^{2}*

Hence: * ^{dx}/_{dt}* = [2(

*E - V] / m*]

^{2}

This equation is important, because the solution for a given E yields x as a function of t. Substituting that value of x into the expression for V(x) yields the particle's velocity at that instant. Knowledge of x(t) and v(t) (or p(t)) enables us to describe the trajectory of the particle, and hence predict its location at any instant! This is the key to the success of classical mechanics.

### Newton's second law – rectilinear motion

For motion in a straight line, this can be written in various forms:

F = ma = m ^{dv}/_{dt} = m ^{d2x}/_{dt2} = ^{dp}/_{dt}

(i.e. the rate of change of momentum equals the force acting on the particle). Hence if we know the force acting on the particle at all times and positions we can obtain its trajectory as above. As an example, consider a particle with initial momentum *p*(0)= 0 (i.e. at rest), subjected to a constant force F for time τ (Greek: “tau”), then allowed to move freely. Then we must have:

*dp/dt = F*(a constant) between* t * = 0 and * t = τ *

*dp/dt* = 0 > t > τ

The solution is p(t) = Ft, for 0 < *t < τ*

Since *K = p ^{2}*/ 2

*m*, then the energy after the force ceases to act is

*E = F*/ 2

^{2}τ^{2}*m*. Note that the particle is initially at rest, and F and τ can assume any value, which implies that (in classical mechanics) the energy of a particle undergoing rectilinear motion may be increased from 0 to any arbitrary value.

### Rotational motion

In this case similar considerations apply, and rotational motion is a very important type of motion in chemistry (can you imagine why?). The concepts of momentum and velocity used above become angular momentum and angular velocity for rotational motion. The angular momentum, J, of a particle is J = Iω (compare this with p = mv) where I is the moment of inertia and ω (Greek: “omega”) the angular velocity. For a point particle of mass m moving in a circle of radius r, I = mr^{2}.

To accelerate a rotation we need to apply a torque, T, or a twisting force, and Newton's equation becomes dJ / dt = T (compare this with dp / dt = F). As for rectilinear motion, application of a constant torque for time τ increases the rotational (kinetic) energy by *E = T ^{2} τ^{2}*/ 2 I. Again, the implication is that an appropriate torque and time interval can excite the rotation to an arbitrary energy. Classical rotational motion is reviewed in Section 18.7 of the textbook.

### The harmonic oscillator

A third type of motion fundamental in chemistry is oscillatory, such as the vibration of atoms in a bond. A harmonic oscillator consists of a particle that experiences a restoring force proportional to its displacement from an origin:

F = -kx

where the negative sign indicates the force is opposite to the displacement, and k is called the force constant. Using Newton's second law we can write:

^{dp}/_{dt} = m^{d2x}/_{dt2} = F = -kx

This is a second-order differential equation, readily solved (check this by substituting the solution into the equation) to yield:

x = Asinωt

p = m^{dx}/_{dt} = mωt

ω = √k / m ⇒ mω^{2} = k

You should be able to see that the position of the particle varies harmonically (or sinusoidally) with a frequency *ν = ω* / 2*π* (ν is Greek “nu”), and is stationary

(*dx/dt* = 0 ⇒ *p* = 0 ⇒ *ωt = nπ*/2

when x has its maximum value of ±A, the amplitude of motion (i.e. at the extremes of its oscillation). The potential energy (yes, it must also possess potential energy, since its kinetic energy is clearly not constant - it is zero at x = ±A!) can be obtained from F = –dV/dx, hence * V *= ½kx^{2}(if *V* = 0 at *x* = 0). Then the total energy can be written:

*E* = ½*p ^{2}/ m + V*

= ½*p ^{2}/ m + *½

*kx*

^{2} = ½*mω ^{2}A^{2}cos^{2}ωt + *½

*kA*

^{2}sin^{2}ωt= ½*kA ^{2}(cos^{2}ωt + sin^{2}ωt)*

= ½*kA ^{2}*

From this we can see that the energy of an oscillating particle can also assume any arbitrary value. Note that the frequency of oscillation depends only on the structure of the oscillator (i.e. k and m) and not on its energy! Amplitude of oscillation determines energy, and it is independent of the frequency. The classical simple harmonic oscillator is reviewed in section 18.6 of the textbook.

### The equipartition theorem

This useful theorem states that the average value of each quadratic term in the energy expression for a particle at temperature T is ½*kT* where k is the Boltzmann constant, k = 1.381 × 10^{-23} JK^{-1}.

(The gas constant, R, is the product of k and N_{A}, R = kN_{A}).

It follows from the equipartition theorem that the average energy of a gas in a container (i.e. undergoing three-dimensional translational motion) is ^{3}/_{2}* kT* (from *E = (p _{x}^{2} + p_{y}^{2} + p_{z}^{2})*/ 2

*m*; the average energy of a collection of one-dimensional harmonic oscillators is kT (from

*E =*½

*p*½

^{2}/m +*kx*), and the average energy of a collection of objects rotating about a single axis is ½

^{2}*kT*(from ½

*Iω*).

^{2}Note that all of these expression for the total energy of a system are expressed in terms of momenta (for a three dimensional system, we have three separate momentum variables, p_{x}, p_{y}, and p_{z}) and position (x,y,z). We will use the same variables when we move from classical to quantum mechanics.

## The failures of classical physics

#### Read Engel and Reid

#### Chapter 12

Two important conclusions emerge from the classical descriptions of motion:

classical physics:

- predicts a precise trajectory, and
- allows all forms of motion to be excited to any energy simply by controlling the forces, torques etc. that may be applied.

It turns out that classical mechanics, which is excellent in describing everyday objects (ball bearings, planets, etc.) fails in describing very small systems and transfers of very small energies. It is only an approximation to the description of matter on the atomic scale. However, its failures (which were spectacular at the turn of the last century) provided the impetus for a completely novel approach to the description of matter, and we now examine some of these examples in turn.

### Blackbody radiation

All bodies emit radiation when heated. As the temperature increases, the apparent colour of the body changes from red to white to blue. In terms of frequency, the emitted radiation goes from a lower frequency to a higher frequency as the temperature increases. Red is in a lower-frequency region of the spectrum (*λ* ≈ 700 nm; *ν* ≈ 4.3 × 10^{14} s^{–1}) than is blue (*λ* ≈ 450 nm; *ν* ≈ 6.7 × 10^{14} s^{-1}). The exact spectrum emitted depends upon the body itself, but an ideal body, which emits and absorbs at all frequencies, is called a blackbody, and its emitted radiation is called blackbody radiation. A good approximation to a black body is a cavity with a tiny hole (see Fig 12.1, p.277 E&R).

The amount of energy radiated by a black body as a function of its temperature is described by the Stefan-Boltzmann law:

M = σT^{4}

where σ (Greek “sigma”) is the Stefan–Boltzmann constant = 5.67 × 10^{-8} J m^{-2} s^{-1} K^{-4} and T is absolute temperature.

As the temperature is raised, more light is emitted at shorter wavelengths (higher frequencies). Frequency (ν) and wavelength (λ, Greek “lambda”) are related to the speed of light (c) by:

λν = c

Classical physics could not explain the temperature dependence of blackbody radiation (Fig 12.2, p.277 E&R).

Many theoretical physicists attempted to derive this distribution function, but were unsuccessful. The best attempt, by Rayleigh and Jeans (equation 12.3, p.277), predicted that the emitted energy by the black body depended on 1 / *λ ^{4}*. This successfully predicted the fall-off in emitted radiation at longer wavelengths (low frequencies). However, it does not predict a maximum but instead predicts increasing emitted intensities at shorter and shorter wavelengths. This failure of classical physics was called the “ultraviolet catastrophe” because it predicted significant radiation in the short wavelength, UV portion of the electromagnetic spectrum. According to this theory objects should glow even at room temperature! Rayleigh and Jeans assumed that the energies of these "oscillators" could have any value at all (a sensible outcome of classical physics as we have seen above) and they used the expression for the average energy of a collection of one-dimensional harmonic oscillators we have given above, E = kT.

Max Planck (1900) was able to resolve the discrepancy between theory and experiment, but it required a rather revolutionary assumption. Like Rayleigh and Jeans, Planck assumed that the radiation emitted was due to the oscillation of electrons in the matter of the body. Planck, however, made the assumption that the energies of the oscillators had to be proportional to an integral multiple of the frequency,

E = nhλ

(12.4)

where h is a proportionality constant. Using statistical thermodynamic arguments, Planck derived equations similar to 12.5 and 12.7 and showed that excellent agreement with experiment could be obtained if *h* = 6.626 x 10^{–34} J s. This constant is of course now known as Planck's constant. For small frequencies it is straightforward to show that Planck's equation reduces to the Rayleigh-Jeans law, but the Planck distribution does not diverge at large ν.

An empirical (what does this word mean?) relationship known as Wien's law was known in the late nineteenth century:

λ_{max}T =2.90 × 10^{-5}

#### Exercise 3.1

Attempt problem 12.1 on page 287 of E&R.

If you are keen, attempt problem 12.12.

From equation 12.7, setting *dρ / dλ* = 0, it can be shown that Planck's distribution predicts

*λ _{max}T = ^{hc}*/

_{4.965k}= 2.90 × m K

The theory of blackbody radiation is not abstract; it is used regularly in astronomy to estimate the surface temperature of stars. For example, from the electromagnetic spectrum of the sun measured in the earth's upper atmosphere *λ*_{max} = 500nm, and from Wien's law we deduce that *T* = (2.90 × 10^{–3} m K)/(500 × 10^{–9} m) = 5800 K

### The photoelectric effect

Examine the Planck distribution as a function of wavelength and frequency at a variety of temperatures:

http://www.oup.com/uk/orc/bin/0198792859/resources/livinggraphs/graphs/P711S01.html

For some time, Planck's derivation was regarded as a curiosity - it was felt that a suitable classical explanation, one that avoided Planck's crucial assumption, would eventually be found. However, in 1905 Einstein used precisely the same idea to explain the photoelectric effect.

The ejection of electrons from the surface of metals when exposed to ultraviolet radiation is known as the photoelectric effect. Its characteristics are:

- no electrons are ejected, regardless of light intensity, unless the frequency exceeds a threshold value characteristic of the metal;
- the kinetic energy of ejected electrons is linearly proportional to the frequency of the incident light, but independent of its intensity;
- even at low light intensities, electrons are ejected immediately if the frequency is above the threshold.

#### Exercise 3.2

Attempt problem 12.11 on page 287 of E&R.

If you are keen, attempt problem 12.20.

The kinetic energy of the ejected electrons can be measured by the potential needed to be applied to a negative electrode to just stop them - the stopping potential *V _{S} *(½

*m*). These observations are summarised in Figure 12.4, p.279 E&R.

_{e}v^{2}= -eV_{S}These observations suggested that the photoelectric effect depends upon the ejected electron being involved in a collision with a particle-like projectile with sufficient kinetic energy to knock it out of the metal. If we suppose the particle is a photon of energy νh, the conservation of energy requires that

½*m _{e}v^{2} = hν - Φ*

Explore the photoelectric effect:

http://www.oup.com/uk/orc/bin/0198792859/resources/livinggraphs/graphs/P711S03.html

where Φ is the work function of the metal - the energy required to remove an electron from the solid. If hν < Φ photoejection cannot occur. This simple relationship predicts that a plot of V_{S>} against ν should be linear, with a slope of –h / e (see Fig. 12.4). Using the known value of e Einstein (in 1905) obtained a value of h in close agreement with Planck's value from blackbody radiation.

### Wave-like behaviour of particles

Scientists have always had trouble describing the nature of light. In some experiments it shows definite wavelike character, but in others it behaves as a stream of particles (now called photons). In 1924 the French scientist Louis de Broglie reasoned that if light displays wave-particle duality, then matter, which is certainly particle-like, may also display wave-like character under certain conditions. Einstein had already shown from relativity theory that the momentum of a photon is given by p = h / λ. De Broglie argued that both light and matter obey the equation

λ = h / p

(12.11)

This equation predicts that a particle with mass m moving at velocity v will have associated with it a wavelength λ = h mv. For example, the de Broglie wavelength of a cricket ball (156 g) travelling at 151.2 kph (40 m s^{–1}) is 1.06 × 10^{–34} m!! On the other hand, the de Broglie wavelength of an electron accelerated from rest through a potential of 1.00 kV is 3.88 × 10^{–11} m = 0.388 Å. The wavelength of such electrons corresponds to the wavelength of X-rays, and like X-rays such electrons are diffracted by crystalline material. This diffraction of electrons was first demonstrated by Davisson and Germer in 1925 and is now commonly used in electron microscopes.

The most dramatic experiment demonstrating the wave nature of electrons is the double-slit diffraction experiment: a beam of electrons fired through an arrangement of two parallel slits will give a diffraction pattern, as though the electrons going through the two different slits could interfere with each other just like classical light beams. This even happens if the electrons are fired through one at a time, suggesting that somehow, a single electron can go through both slits simultaneously! It is possible to carry out experiments to find out which of two slits an electron went through... but if this is done, the interference patterns disappear. Subsequent experiments have shown that other particles, such as protons, neutrons, and even helium nuclei also exhibit this bizarre behaviour.

The wave-particle duality is not obvious from our usual macroscopic observations. Most macroscopic particles have significant masses such that their de Broglie wavelength is infinitesimally small. Microscopic particles, such as photons and electrons are neither waves nor particles, but something else. They can appear to behave as waves or particles, depending on what sort of experiment we carry out. An accurate pictorial description of these particles’ behaviour is impossible using only the wave or only the particle concept of classical physics. The concepts of classical physics have been developed from experience in the macroscopic world and do not provide a proper description of the microscopic world.

Although both photons and electrons show an apparent duality, they are not the same kinds of entities. Photons always travel at speed c and have zero rest mass; electrons always have ν < c and a nonzero rest mass. Photons must always be treated relativistically, but electrons whose speed is not too high can be treated nonrelativistically.

Another important consequence of the wave nature of particles is what is called the uncertainty principle. The momentum of a particle is a function of its wavelength (12.11), but it is impossible to specify an exact wavelength for a localised wave: if a wave packet occupies only a particular bit of space, it must necessarily be composed of a sum of waves with a number of different wavelengths. Thus, if a particle actually is somewhere, rather than everywhere, there will be some uncertainty in its wavelength, Δ*λ*. In the same way, unless it is described by the sum of an infinite number of waves, there will be some uncertainty in its position, Δx. It is possible to put numbers on this trade-off between momentum and position, which is done much later on in the textbook (pp.360-366) to give the expression

Δ*p*Δ*x* ≥ *h / *2

#### Exercise 3.3

The following summary by the famous physicist Richard P. Feynman (The Feynman Lectures in Physics, III-1.10) defines an important term which we will meet again. [An ‘event’ is, for instance, an electron arriving at a specific point on a detector. The ‘several alternative ways’ are, for instance, electrons passing through slit 1 or slit 2.]

- The probability of an event happening in an ideal experiment is given by the square of the absolute value of a complex number Φ which is called the probability amplitude:
- When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference:
- If an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. The interference is lost:

P = probability, Φ = probability amplitude P = |Φ |^{2}

Φ = Φ_{1} + Φ_{2}

P = |Φ_{1} + Φ_{1} + Φ_{2}|^{2}

P = P_{1} + P_{2}

P_{1} = |&Phi_{ 1}|^{2}

P_{2} = |Φ _{2}|^{2}

[Pick any function you like for Φ_{1} and Φ_{2} and create plots illustrating cases (2) and (3) for those functions.]

### Atomic Spectra

A detailed analysis of the emission spectrum of hydrogen was a major step in the elucidation of the electronic structure of atoms. In 1885 a Swiss amateur scientist, Johann Balmer, showed that a plot of the frequencies of the lines in the visible/UV spectrum versus 1/n^{2} is linear. In particular Balmer showed that

*ν* = 8.2022 × 10^{14}(1 - ^{4}/_{n2}) Hz (Hz = s^{-1})

where *n* = 3,4,5,... etc. It is customary now to write this in terms of 1 / *λ* instead of n. Reciprocal wavelength is denoted in the text by ṽ, measured in the standard unit cm^{–1} (these are non-SI units, and are called wavenumbers). We can then obtain the Balmer formula:

*ṽ* = ^{1}/_{λ} =* R*(^{1}/_{22} - ^{1}/_{n2}),* n* = 3,4,5,...

There are series of lines just like the Balmer series in the UV and infrared regions of the spectrum. The Swiss spectroscopist Johannes Rydberg accounted for all the lines in the H atomic spectrum by generalising the Balmer formula:

*ṽ* = R_{H}(^{1}/_{n22} - ^{1}/_{n21})

(12.13)

#### Exercise 3.4

Attempt problem 12.17 on page 288 of E&R.

This is the Rydberg formula and the constant occurring in it is now called the Rydberg constant, RH = 109,677.57 cm^{–1}.

The fact that the formula describing the entire H atom spectrum is controlled by two integers is quite surprising. But integers play a special role in quantum theory, as we shall see.

# Topic 3B – Introduction to Quantum Mechanics

In 1926 both Schrödinger and Heisenberg independently formulated a general quantum theory - a new kind of mechanics - capable of dealing with the wave-particle duality of matter. They used quite different mathematical formalisms but both represent different forms of what is now known as quantum mechanics. Schrödinger's mathematics (wave mechanics) involves differential equations and is more familiar to chemists; it is usual to use his equations as the basis of chemical applications of quantum mechanics - quantum chemistry.

## When do we use Quantum Mechanics?

Read Engel and Reid

Chapter 13.1

This section attempts to give a more quantitative answer to the question ‘when do we use quantum mechanics?’ than, ‘when we’re talking about really small things’. Crucial to answering this question is the Boltzmann distribution, which you may possibly remember from first year. It is important that you learn this important classical expression for the partitioning of particles (or oscillators, or any collection of systems) between two energy states separated by an energy difference Δ*E*:

^{ni}/_{nj} = ^{gi}/_{gj} e ^{-[εi - &epsilonj]/kT}

(13.2)

where n_{i} is the numbers of atoms that have energy &epsilon_{i} and n_{j} is the number of atoms that have energy ε_{j}, g_{i} and g_{j} are the degeneracies of the energy levels- the number of possible ways of arranging the system to get to that energy. k is Boltzmann’s constant (8.314 JK–1 / Avogadro’s Number) and T is the temperature.

Basically, if ε_{i} - ε_{j} << kT, then quantum mechanical behaviour will be observed: otherwise, the system will display a more-or-less continuous distribution of energy levels and behave more-or-less classically.

Read Engel and Reid

Chapter 13.2 and 13.3

### Classical Waves

Consider a uniform string stretched between two fixed points (e.g. a string on a violin). The displacement of the string from its equilibrium horizontal position can be described in terms of a wavefunction ψ(x,t): the amplitude of the displacement as a function of both position x and time t. Using classical mechanics it can be shown that ψ(x,t) satisfies the equation

^{d2ψ}/_{dx2} = ^{1}/_{v2} ^{d2ψ}/_{dt2}(13.11)

where is the speed with which the disturbance moves along the string and the other quantities are partial derivatives. This is called the classical non-dispersive wave equation. Solutions to this equation can be written as trigonometric functions, or as complex functions- you might remember Euler’s equation, e^{ia} = cosa + isina (if only for the gee-whiz expression, e^{ip} = –1).

#### Exercise 3.5

It can readily be shown that a sine wave, travelling in the x direction with velocity v, wavelength λ, frequency **ν** and amplitude A, described mathematically by

*ψ(x,t) = A sin*^{2π}*/ _{λ<} (x - vt) = A sin *2

*π(x / λ -*)

**ν**tis a solution to the wave equation 13.11. Prove this for yourself by substitution and differentiation – but note the difference in this equation between velocity v (which appears in the x – vt term) and frequency **ν** (which appears in the x/λ – **ν**t term).

Read Engel and Reid

Chapter 13.2 and 13.3

Many problems in physics and chemistry are independent of time (e.g. the electron distribution or the energy of a molecule), and we can actually obtain a wave equation that does not contain t as a variable. In mathematics this procedure is known as separation of variables, and it is performed by assuming the solution ψ(x,t) can be written as the product of two functions, one of which depends only on x, and the other only on t:

ψ(x,t) = ψ(x)φ(t)

Substituting this form of ψ (x,t) into the classical wave equation results in:

Left hand side: ( ^{∂2ψ}/_{∂x2} )_{t} = φ(t)( ^{d2ψ(x)}/_{dx2} )

Right hand side: ^{1}/_{v2}* ( ^{∂2ψ}/_{∂t2} )_{x}* =

^{1}/

_{v2}

*ψ(x) (*)

^{d2φ(t)}/_{dt2}dividing both sides by ψ(x)φ(t) we obtain

^{1}/_{ψ(x)}* ^{d2ψ(x)}/_{dx2}* =

^{1}/

_{v2}^{1}/

_{&phi(t)}^{d2&phi(t)}/_{dt2}*function of x only function of t only*

Since x and t are independent variables, the only way a function of x and a function of t can be equal for all x and t is if they both equal a constant, say, -β^{2} (this curious choice will become clearer below!). Then equating each side of the equation to -β^{2} results in two equations, one in x and the other in t.