From Classical to Quantum

Objectives

In this notebook, you learn how to use Wolfram Alpha to explore failures of classical mechanics and the rise of quantum mechanics.

Constants

In these exercises, you will need to use the speed of light, Planck’s constant, and Boltzmann’s constant. Use Wolfram/Mathematica to enter these constants. You may use this document as the template for your work.

Classical Limits

After James Clerk Maxwell derived the equations for electromagnetism, scientists believed that they had developed the theoretical framework for all physical phenomena. The prevailing view was that little was left but to further refine experiments to more precisely describe natural phenomena. A closer look, however, generated results in many cases that were inconsistent with classical predictions. At the beginning of the

century, these strange results led to the development of revolutionary theories that introduced paradoxical behavior at the microscopic level. In the sections below, we describe the observations that led to the development of theories of quantization and wave-particle duality. We will do exercises corresponding to each of the following:

◼

Blackbody Radiation

◼

Photoelectric Effect

In the exercises below (indicated by bullet points), use the Mathematica functions. Do not use free-form entry. The objective, here, is for you to learn the basic functions and elementary coding.

Blackbody Radiation

Background

Blackbody Radiation? What is that? Well, if you consider a spherical cow...A blackbody is defined as an ideal emitter and absorber of electromagnetic radiation. An ideal blackbody might be an object with an interior cavity with a pinhole through which radiation may escape. The walls of the interior cavity absorb all radiation equally and possesses an emission spectrum that is characteristic of its temperature. Who cares, you say? Think about what happens when you put iron in a fire. As it heats, you will note that it first glows red, then, orange, then yellow until it is finally white hot--at white hot temperatures iron will eventually melt. Recognizing this trend, we see that the color shifts to longer wavelengths as the temperature increases. Simple enough, right? The problem is that, while this trend was readily observable, scientists at the end of the 19th century could not explain it. Quantitative observations provided scientists with empirical equations that related the emission output to the wavelength or frequency of light, but classical mechanics did not provide a rationale for these expressions.

Planck Expression

Since classical theory did not yield results consistent with observation, a new theory was necessary. German physicist Max Planck began work on this problem in the 1890s. In December 1900, he derived an expression that fit the data, but he needed to develop an explanation.

He proposed, contrary to classical wave theory, that the electromagnetic emission could only occur with energies proportional to the frequency of the radiation.

E = nhv

The corresponding expression for energy density is

[ν,T]=

8πh

⋁



h⋁



-1

Exercise 1: Plot the Planck Expression

◼

Enter the Planck expression for energy density as a Mathematica function. You will need to enter values for the fundamental constants. Because this plot won’t work with units, enter the SI values of these constants without units.

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Plot the Planck equation. (The plot doesn't work with units.) Make sure all constant values are consistent with SI Units.

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Plot the Planck equation for 300 K, 500, and 1000 K. Plot all on the same graph.

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Label the axes.

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Include a legend.

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Describe the observed trend.

Exercise 2: Frequency Maximum

Find the frequency corresponding to the energy density maximum for each of these temperatures.

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First, estimate the frequency maximum based on your plots above.

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Now, how do you determine the maximum of a function?

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To determine an accurate value, there are several Mathematica functions that work. Two functions that are effective in this case are FindRoot and NMaximize. Try the FindRoot function first. I'll leave it up to you to figure out how to use it.

Exercise 3: Total Energy Density

◼

Determine the total energy density at each of these temperatures in the visible region of the spectrum.

◼

Describe the trend.

Quantum Mechanics

The failures of classical mechanics led to the development of quantum mechanics. For information on fundamentals, see other Wolfram notebooks. In this exercise, we explore Boltzmann population distributions and wavefunction superpositions.

Exercise 1: Level Populations & Temperature

The relative populations of energy levels depends on temperature and level degeneracy. Ludwig Boltzmann developed a statistical approach to predict physical properties of macroscopic systems. To predict level populations in a stable (equilibrium) system, we use the Boltzmann equation:

-ΔE/(kT)

This equation allows you to compute the ratio of particles of level j relative to level i. In this equation, N represents the number of particles occupying a specific level and g represents the corresponding degeneracy. ΔE is the energy difference between the two levels, T is the temperature in Kelvins, and k is the Boltzmann constant.In the problem below, we compute the ratio of molecules in the two lowest vibrational levels in molecular

. The difference between level 0 and level 1 is 2330

-1

◼

Convert 2330

-1

to wavelength.

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Convert the wavelength value to ΔE for the difference between levels.

◼

The vibrational levels in

are nondegenerate. Compute the ratio

at 300 K and at 5000 K.

◼

What happens to the ratio as the temperature increases?

Exercise 2: Superpositions

A superposition is a linear sum of other mathematical functions. Indeed, all mathematical functions may be re-written as a linear combination of other functions. In quantum mechanics, we frequently re-write the state wavefunction as a superposition of eigenfunctions of a specific operator (one that corresponds to a specific measurement). This allow us to determine the probability of measuring specific outcomes in the lab. In general, we represent superpositions mathematically as

ψ=

+...

or, more concisely

ψ=

∑

(

)

The

coefficients are critical for the determination of outcome probabilities as we’ll see in a bit. So, let’s deal specifically with momentum values associated with the particle-in-a-box wavefunction. Wait, how can we write a sine function as a linear combination of imaginary exponentials,

ikx

. Well it turns out there’s a nice little equation called Euler’s formula that helps us with this transformation. For a sine function, we get

Sin[nπx/a]=

-i(nπx/a)



i(nπx/a)



(

)

The exponential terms on the right side of the equation are eigenfunctions of the momentum operator, meaning that there are two possible outcomes to a momentum measurement (because there are two functions in the superposition) and the two outcomes are

-n π ℏ/a and n π ℏ/a

So, in this manner, we can determine measurement outcomes with the appropriate superpositions.

Ok, so that was a short story made long. What exactly are you supposed to here? (Learn to use the Sum function to construct the superpositions below.)

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Start by writing the wavefunction for the particle-in-a-box as a function of x and n. Set the box length a = 1.

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Now, create a superposition of two wavefunctions (n=1 and n=2). Graph the corresponding probability density from 0 to 1.

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Repeat with a superposition of five wavefunctions (choose 5 different n-values). Plot the function.

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Repeat with a superposition of 20 wavefunctions. Plot the resulting function.

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What happens to the function as you increase the number of functions in the superposition?

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What does this indicate about the uncertainty in position? Does the uncertainty increase or decrease as you increase the number of functions in the superposition?

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What does this indicate about the uncertainty in momentum? Does the uncertainty increase or decrease as you increase the number of functions in the superposition?