PHYSICAL CHEMISTRY IICHEM 3354

• Topics: Quantum mechanics, spectroscopy, and statistical mechanics

• Grading: Three exams worth 75%, ACS certified comprehensive (over all p.chem) final worth 25%.

• You are strongly encouraged to read “In Search of Schrodinger’s Cat” by John Gribbon.

• Classical mechanics was introduced in the 17th century by Isaac Newton.– Successful at explaining the motion of everyday objects and planets– Fails when applied to very small particles.– The failures of classical physics lead to the discovery of quantum

mechanics.

• Quantum mechanics was discovered in the 1920s to describe very small particles.– The basis of quantum mechanics is that energy is quantized – has

discreet values.– Particles have wave-like characteristics.

The Origins of Quantum Mechanics

Black Body Radiation

• Hot objects emit electromagnetic radiation.– Example: Hot objects glow “red hot”.

– Hotter objects glow “white hot”.

• We’ll soon see that we need quantum mechanics to describe this phenomena.

Black body radiation

• These are the curves from an ideal black body.– An ideal blackbody is an object capable of emitting and absorbing

all frequencies of radiation uniformly.

• At lower temperature the wavelength with maximum intensity is toward the red end of the spectrum.

• As T increases, the wavelength of maximum intensity blue shifts.– But there’s still a lot of intensity near the red end of the spectrum.– This is why hotter things appears white.

Ideal Black Body

• A pinhole in an enclosed container.– Any emitted radiation will be reflected MANY times before

leaving through the hole.

– The emitted light will consequently be in thermal equilibrium with the wall at temperature T.

Wien Displacement law

• Predicts the wavelength of maximum emission at a temperature T.

• Based only on experimental observation. (i.e. empirical)

Stefan-Boltzmann Law

• Describes the total energy density (the area under the curve)– Total energy density - the

total electromagnetic energy in a region divided by the volume.

– Stefan derived the formula from experimental observations.

– Boltzmann showed the formula theoretically.

Rayleigh-Jeans Law

• Nineteenth century scientists had a difficult time explaining black-body radiation.

• Lord Rayleigh & James Jeans derived from the laws of classical physics the Rayleigh-Jeans law:– The electromagnetic field was

believed to be a collection of oscillators of all possible frequencies.

– The presence of radiation of frequency signified that the oscillator of that frequency had been excited.

– Average energy of each oscillator was kT from equipartition principle. Electromagnetic vacuum supports

oscillations of the EM field

Rayleigh-Jeans law

• According to classical physics, even cool objects should radiate in the visibile and UV regions.– In other words, objects should

glow in the dark.

– Wein and Stefan-Boltzmann laws were not predicted.

• This break-down of classical physics is known as the UV catastrophe – first observed failure of classical mechanics.

• Why did it fail? – Rayleigh assumed that the

electromagnetic energy varied uniformly.

Planck’s Distribution

• Planck proposed that the electromagnetic energy was limited to discrete values, quantization of energy.

• Applying this assumption and thermodyanamics, Planck derived the Planck Distribution:

Planck’s Distribution

• Matches experimental data by predicting that cold object will not emit high energy light.

• First evidence that energy is quantized.

• Planck’s distribution could also account for Stefan-Boltzmann and Wien laws.

Energy Quantization and Thermal Properties of Solids

• It was shown by Einstein that energy quantization must be introduced in order to explain the thermal properties (heat capacity) of solids.

Correct shapePoor fit to data

Debye improved on Einstein’s work

Atomic Spectra and Energy Quantization

• Atomic and molecular spectra provide the most compelling evidence for the quantization of energy.– Energy is discarded or absorbed only in discrete amounts.

Atomic Spectra and Energy QuantizationExplanation of Spectral Lines

• Spectral lines appear because the molecule emits a photon as it changes between discrete energy levels.– High-frequecy radiation is

emitted when the energy change is large.

– E = h

CONCLUSION

Energy is quantized.

Wave-particle Duality

• Classical physics treats electromagnetic radiation as a wave.

• Classical physics treats electrons as particles.

• Let’s look at some experiments that invalidate these ideas.

The Particle Character of EM RadiationPhoto-electric effect

• Electrons are ejected from metals when exposed to ultraviolet radiation.

• Experimental Observations:1. No electrons are ejected, regardless

of intensity, unless its frequency exceeds a threshold value characteristic of the metal.

2. The KE of the ejected electrons increases linearly with the frequency of the incident radiation but is independent of the intensity of the radiation.

3. Even low intensities immediately eject electrons if the frequency is above threshold.

• Conclusion: the ejection of an electron occurs due to a collision with a particle-like projectile that carries enough energy to eject the electron from the metal.

Photon energy is insufficient

Photon energy is sufficient and excess energy is carried away as KE of the photoelectron

Photo-electric effect

• Light behaves as a particle, called photons, each photon has an energy

hc

hvE

Davisson-Germer Experiment

• The scattering of an electron beam from a nickel crystal showed a variation of intensity.– This variation is characteristic of

a diffraction experiment in which waves interfere constructively and destructively in different directions.

• The experiment has been repeated with heavier particles (including molecular hydrogen) and shows clearly that particles have wave-like properties.

• Electron diffraction the basis of electron microscopy.– Analogous to optical microscopy

but the image is constructed from the pattern of scattering of a focused beam of electrons.

The de Broglie wavelength

• De Broglie proposed that any moving particle (with a linear momentum ) should a wavelength

– These “fictitious waves” are called mater waves.

– The de Broglie relation can be derived from Eistein’s relativity theory and the Planck-Einstein relation.

– The de Broglie relation can also be derived by creating a standing wave on a ring (i.e. Bohr atom and quantization of angular momentum).

• Exercise: Calculate the de Broglie wavelength of a baseball of mass 5.1 ounces thrown at 59 miles/hour.

– Macroscopic bodies have such high momenta (due to large m) that their wavelengths are undetectably small.

Conclusion

Matter and light can behave as particles or waves.

A wave-particle duality exists.

Classical physics treats particles and waves as separate entities. Must have a new

description to account for this mysterious behavior.

Brace yourselves for quantum mechanics!!!!

Units and constants in Quantum Mechanics• Scientists have developed a set of energy units that are

convenient for describing atomic and molecular energies.• Energy Units

– Electron volt = eV = 1.602 x 10-19 J {electron times a volt}– Hartree = Eh = 27.2 eV– Wavenumber = cm-1 with 1 eV = 8065.5 cm-1; 1 cm-1 = 1.9864 J

• Commonly used constants.– Speed of light = c = 2.998 x 108 m/s– Planck’s constant = h = 6.626 x 10–34 J s– Atomic mass unit = u = 1.66054 x 10-27 kg– Angstrom = Å = 10-10 m– Bohr = ao = 0.529 Å

Classical Wave TheoryPhysics Review

• Consider the harmonic-motion of a one-dimensional string (i.e. a guitar string)– The classical time-independent wave equation is:

– The solution in 1-D is: 3-D:

– Apply results to particle with a de Broglie wavelength .

Time Independent Schrödinger Wave Equation (SWE)

• This is postulate of QM, accounts for wave character of matter.

• The wavefunction contains all the dynamical information about the system it describes.– The trick is to determine what is.

– And to figure out how to extract the desired information.

Vm

H

EH

22

2ˆ

ˆ

Interpreting the wavefunction Born interpretation of

• According to the wave theory of light, the square of the amplitude of an EM wave is proportional to the intensity of light.– But since light behaves as a particle, the intensity must be a measure of

the probability density of photons in a volume of space.

• Applying this same idea to particles indicates that the value of | at a point is proportional to the probability of finding the particle at that point.

Born Interpretation of • The probability of finding a

particle between x and x+dx is proportional to |dx.– |probability density (real

and never negative).

– probability amplitude• Can be complex

In 3-D, the probability of finding a particle in an infinitesimal volume d = dxdydz is

proportional to ||2 d

The Born interpretation places some restrictions on the form of the wavefunction.

1. must be finite everywhere; otherwise couldn’t be normalized.

2. must be single valued; otherwise could have more than one probability.

3. must be defined everywhere.4. The second derivative of must be well-

defined; or the SWE will not be applicable everywhere.

5. These restrictions prevent the possibility of finding an acceptable solution to the SWE for arbitrary values of E.– Only certain energies (quantization) have

acceptable solutions.

Not continuous

Slope is discontinuous

Not single-valued.

Infinite over a finite region.

The Born Interpretation and Normalization

• The probability that a particle is somewhere must be 1.– For a normalized wave function in 1-D:

– If the wavefunction is NOT normalized, we multiply by a constant factor N to normalize it.

1)()(* dxxx

1*2 dN

Normalization Example

• The wavefunction for an electron in the lowest energy state of a hydrogen atom is proportional to e-2r/ao where ao is a constant (the bohr radius) and r is the distance the electron is from the nucleus. Normalize this wavefunction.

Spherical Coordinates

How do we get other information (besides position) out of the wavefunction?

. fromenergy theextracts -operator energy operator,n Hamiltoniaˆ

ˆ

H

EH

• The Schrodinger equation is an eigenvalue equation!(operator)(eigenfunction) = (eigenvalue)(same eigenfunction)

• Examples – Are these functions eigenfunctions of the d/dx operator? If so, what are

the eigenvalues.

– Is the function cos (ax) an eigenfunction of:

2ax

ax

e

e

2

2

dxd

dxd

To ‘solve the Schrodinger equation’ is to ‘find the eigenvalues and eigenfunctions of the Hamiltonian operator for the system of interest.’

•The wavefunctions are the eigenfunctions of Ĥ.

•The corresponding eigenvalues are the allowed energies.

Let’s solve it…

• Solve the SWE for a particle with mass m free to move on the x-axis w/ zero potential energy (V = 0).– cos(kx) to is an eigenfunction of the SWE (and consequently a valid

solution) with an eigenvalue.

• What is k?– It’s the “quantum number”. BUT since the solution is valid for ANY

value of k the energy of this particle is NOT quantized.– k controls the frequency of the cosine wave.

• Large k high momentum many oscillations wiggly wavefunction large curvature.

• Small k low momentum few oscillations smoother wavefunction small curvature.

kpm

p

m

kE

;

22

222

The Momentum Operator

• There’s a QM operator for all classical mechanical observables.– Ĥ is the energy operator extracts the energy from .

• Finding the momentum from this energy only gives its magnitude, not its direction.

– The momentum operator is.

• Problem: Doesn’t work with cos (kx), not an eigenvalue equation.

dx

d

ipx

ˆ

kpm

p

m

kE

;

22

222

• Let’s represent cos(kx) in terms of exponential functions, a more flexible solution.

• A wavefunction that is not an eigenfunction of the operator of interest can be written as a linear combination of eigenfunctions.

• Conclusion: The wavefunction is a superposition of states (moving EITHER left OR right with momentum ħk). – We don’t know which state our system is in until we observe it.– Since both directions are equally probable, we cannot predict the direction the

particle will be traveling.– Introduce the Schrodinger cat paradox.

Solution

Schrödinger’s Cat Paradox

• Cat is in closed container containing a vial of hydrocyanic acid.

• One atom of a radioactive substance (half life = 1 hour) is in a device with a Geiger counter.

• If the Geiger counter detects a decay, it triggers the release a hammer to fall and break the vial—killing the cat.

• Since the probability of decay is 50/50, there is a 50/50 chance the cat will be alive after 1 hour.

• The cat is both dead and alive until it is observed.

For all classical mechanical observables, there’s a corresponding quantum mechanical observable.

For all classical mechanical observables, there’s a corresponding quantum mechanical observable.

• If is an eigenfunction of our quantum mechanical operator, then we can operate on to obtain our classical mechanical observable.

• What if is NOT an eigenfunction of our operator?– Then we can calculate an expectation value of this operator.

• An expectation value is a weighted average of a large # of observations of a property.

.normalized bemust :NOTE

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dOO

Example expectation value problem

• Calculate the average value of the distance of an electron from the nucleus in the hydrogen atom.

PROOF

• A wavefunction that is not an eigenfunction of the operator of interest can be written as a linear combination of eigenfunctions.– Example: cos kx was not an eigenfunction of the momentum operator

but eikx + e-ikx was.

– Find <Ô> for = c11 + c22 where 1 and 2 are eigenfunction of Ô.

• The expectation value is the sum of the eigenvalues weighted by the probabilities that each one will be found in a series of measurements.

Heisenberg’s uncertainty principle

• For = Aeikx we found that the particle was moving to the right with px = +kħ. – Therefore, we know its momentum exactly.

• Where is the particle? (probability of finding the particle at a location x)– Probability = * = A2 e-ikx eikx = A2

– There is an equal probability of finding the particle anywhere on the x-axis.

• Conclusion:– We know the momentum of the particle exactly.

– BUT, we know NOTHING about the location of the particle.

• It is impossible to specify simultaneously both the momentum and position of a particle.

Heisenberg’s uncertainty principle

• If we know a particle is in a definite location, then must be large there and zero everywhere else.

Heisenberg’s uncertainty principle

• To make such a wavefunction (and still have a solution to the SWE) it must be a linear combination of many harmonic (sine or cosine) functions, or equivalent, eikx functions.– It takes an infinite # of these to

make an infinitely narrow spike.

– But each eikx function corresponds to a different linear momenta.

– Therefore there’s an infinite number of momenta possibilities.

• It is impossible to specify simultaneously both the momentum and position of a particle.

Heisenberg’s uncertainty principle

• The quantitative interpretation of the uncertainty principle is:

2 xpx

Uncertainty in the momentum

Uncertainty in the position

22

22

xxx

ppp

p and x are the ‘uncertainties’ in the momentum and position, respectively.

As either x or px approach zero, the other observable must approach infinity.

Heisenberg’s uncertainty principle and operators

• Heisenberg’s uncertainty principle applies to any pair of operators  and B that do NOT commute.– The commutator of two operators is defined by:

– If [Â ,B] = 0, then the operators commute and CAN be measured simultaneously.

• Find the commutator of px and x.

• Find the commutator of px and Ť. – What does this tell you about these two observables?

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