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# Classical vs. Quantum Mechanics

Classical mechanics consists of the work done in the areas of chemistry and physics prior to the 20th century. This includes the organization of the periodic table, thermodynamics, the wave theory of light, and Newtonian mechanics. Quantum mechanics was born out of the inability of classical mechanics to reconcile theory with experiment.

### Introduction

Some of the areas in which discrepancies were observed between the classical model and experiment are: blackbody radiation, the photoelectric effect, and heat capacity.

Blackbody radiation provided a famous incongruity between theory and experiment for classical physics. Classically, the radiant energy density $$d\rho$$ was described by the following equation:

$d \rho ( \nu , T) = \frac{8 \pi \kappa_B T}{c^3} \nu^2 d \nu$

This led to the problem known as the "UV catastrophe." As the frequency of light $$\nu$$ increases the radiant energy density approaches infinity (Figure 1). However, this trend was not observed in the experiments. In fact, the radiant energy density was found to decrease as the frequency increased in the UV spectrum.

In 1900, Max Planck successfully explained blackbody radiation and derived the following equation to accurately describe the experimental results:

$\large d \rho ( \nu, T)= \dfrac{8 \pi h}{c^3} \frac{\nu^3 d \nu}{e^{\frac{h \nu}{\kappa_B T}}-1}$

Planck was able to derive this formula by assuming that the energies of the oscillators were quantized (i.e. $$E = n h \nu$$,  where $$h$$ is Planck's constant $$h=6.626*10^{-34} Js$$. Planck's quantization of energy was a revolutionary assumption that marked the beginning of a new field of chemistry aptly named quantum mechanics.

Figure 1: Line plot depicting the classical model of blackbody radiation overlapped with the quantum model of blackbody radiation. The quantum model agrees with the experimental values whereas the classical model diverges at high frequencies. This divergence is what is referred to as the UV catastrophe.

Notice that as the frequency of the radiation decreases the quantum prediction approaches the classical prediction.

### Photoelectric effect

Another phenomenon in which a quantized approach was used to explain the experimental results is the photoelectric effect. Classical physics describes light as a wave (electromagnetic radiation) with a set frequency and amplitude where the amplitude is related to the intensity.  Light was observed to cause electrons to be ejected from a metal's surface. The classical explanation was that the metal's electrons would oscillate with the light and eventually break away from the surface with a kinetic energy that would depend on the intensity of the incident radiation because the higher the intensity the higher the amplitude of the oscillation. However, the kinetic energy of the ejected electrons was shown to be independent of the intensity of the radiation. In fact, there were some frequencies that no matter how intense the incident radiation was no electrons were ejected.

Einstein modified Planck's concept of quantized energy to describe the experimental results. Einstein proposed that light could travel in small quantized packets of energy (photons) instead of strictly behaving as a classical wave. Einstein showed that the kinetic energy of the ejected electrons was equal to the energy of the incident photon ($$h \nu$$) minus the energy barrier to releasing an electron from that particular metal (workfunction=$$\phi$$).  This interpretation is described by the following equation:

$$KE= \frac{1}{2} m v^2 = h \nu - \phi$$

This model was able to fully account for the experimental results including the lack of dependence of the energy of the ejected photons on the intensity of the incident radiation as well as the failure of some frequencies of light to eject any photons (the incident energy of the photon was less than the workfunction).

One important result of Einstein's work with the photoelectric effect (outside of the concept of the photon) was the fact that his experimentally determined value of $$h$$ was the same value determined by Planck. This gave credence to the idea of quantized energy and quantum chemistry as a whole, which was still viewed with suspicion by many scientists.

### Molar heat capacity at constant volume ($$C_v$$)

Classically, the molar heat capacity at constant volume is equal to $$3R$$ where $$R$$ is the molar gas constant ($$8.314 J K^{-1} mol^{-1}$$). This model was found to hold true at high temperatures, but to break down as the temperature is decreased. Einstein relied on a quantum interpretation to explain why the experimental results deviated from the classical prediction at low temperatures.

Einstein proposed that the physical oscillations of the atoms in the crystal matrix are subject to quantized vibrational states where the change of position of the atom in the lattice ($$\Delta \varepsilon$$) is represented by the equation:

$$\Delta \varepsilon = h \nu$$

Figure 2: Line plot of the classical and quantum models for heat capacity at constant volume.

Notice that as the temperature increases, the quantum model approaches the classical model. This was an interesting addition to quantum theory because it extended the concepts beyond light and electrons to the mechanical vibrations of whole atoms.

### Conclusion

Classical mechanics accurately describes most systems that can be easily observed.  Objects that are a "normal" size (larger than a molecule and smaller than a planet), at a "normal" temperature (anywhere close to room temperature), going a normal speed (0 m/s- anything significantly less than the speed of light) fit the models set forth in classical mechanics.  It is only when the system being observed begins to violate these parameters that quantum factors come into play.  An important aspect of the quantum mechanical models is the fact that as the conditions approach "normal" the quantum mechanical model approaches the classical model.

### Summary

Quantum mechanics approaches Classical mechanics when:

• $$\nu \rightarrow 0$$: This is observed in the phenomenon of blackbody radiation (Figure 1).
• $$t \rightarrow \infty$$: This is observed in the phenomenon of heat capacity (Figure 2).
• $$h \rightarrow 0$$: This is observed when taking the limit as $$h \rightarrow 0$$ for the average quantum mechanical energy ($$\frac{h \nu}{e^{\frac{h \nu}{k_B T}} -1}$$).  Notice that this limit is equal to the average classical energy ($$k_B T$$).
• $$n \rightarrow \infty$$: This is known as the Bohr correspondence (Figure 3).

Figure 3: Line graph of the probability of finding a particle at a given position for two wave equations evaluated at different $$n$$'s.

Notice that as $$n$$ increases, the probability function approaches a straight line. Classically, the probabilty of finding a particle is independent of position (a straight line with a constant y value).  Thus, as $$n$$ increases, the quantum model approaches the classical model.  This is known as the Bohr correspondence.

### References

1. McQuarrie, Donald A. Quantum Chemistry. 2nd ed. United States Of America: University Science Books, 2008.

### Problems

1. Prove that taking the limit as $$h \rightarrow 0$$ for the average quantum mechanical energy ($$\frac{h \nu}{e^{\frac{h \nu}{k_B T}} -1}$$) yields the average classical energy ($$k_B T$$).  (Hint: use l'Hôpital's rule)

14:38, 17 Sep 2014

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This material is based upon work supported by the National Science Foundation under Grant Numbers 1246120, 1525057, and 1413739.