Wavefunctions

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Wave Functions

Introduction

Quantum mechanics was a field pioneered by theoretical physicists such as Louis de Broglie, Erwin Schrödinger, Werner Heisenberg, and Wolfgang Pauli. De Broglie and Schrödinger used the theories of calculus in wave mechanics, whereas Heisenberg and Pauli used the theories of linear algebra to describe what they called matrix mechanics. Schrödinger eventually proved that both approaches were the same; the wave function unified the two.

The wave function expresses how a particle behaves in time and space in quantum mechanics.

Classical Waves

The study of classical waves began with mechanical waves, which travel through a medium (e.g. air, water, a material). Electromagnetic waves, on the other hand, have electric and magnetic fields undulating perpendicular to each other and the direction of propagation and can travel through a vacuum, meaning it doesn't require a medium.

Both of these are traveling waves and move through space at speed c. In classical mechanics standing waves are also studied. These are waves that have fixed ends. The wave then can only move back and forth between its two fixed ends, just like strings on guitars and violins.

Continuing with traveling waves, superposition is when two waves travel through the same space at the same time. This can cause two kinds of interference: constructive or destructive. If the crests of each wave are aligned then their amplitude is the sum of both waves, and this is constructive interference. If the crest of one aligns with the trough of the other, then their amplitudes cancel out. That is destructive interference. If the two waves do not perfectly align then their amplitude is the intermediate between the constructive and destructive values. The superposition principle can be applied to standing waves in the same form; they can be given in terms of a sum or superposition of traveling waves (described as superimposed traveling waves). It also includes that the basis of all traveling waves can predict wave function dynamics.

Classical waves are described mathematically with many equations.

c= λv

where c is speed, λ is wavelength, and v is frequency.

An example of a complex wave function is the complex representation of a single traveling wave:

cos(kx-wt+φ)=Re{e^i(kx-wt+^φ)}

where k is the wave number (k= 2π/λ m^-1), w is the angular frequency (w= 2π/T s^-1 (T is the period)), and φ is the phase of the wave.

Classical waves are described by amplitude, wavelength, and frequency. They are separate from particles and their exact positions can be calculated. When you approach the classical limit of waves, you reach Einstein’s wave-particle duality.

Wave-Particle Duality

Wave-particle duality was discovered as a consequence of the photoelectric effect. De Broglie related standing waves to the wave property of electrons. It is important to understand that electrons act as particles and waves to understand the meaning of wave functions.

Wave Functions

“What is the physical meaning of the wave function?”

The wave function describes the state of the system and is a mathematical tool to calculate probabilities and probability densities. The probability density for finding the particle at position x is:

Where psi star is the complex conjugate of psi. The complex conjugate is used to eliminate any complex numbers that exist in psi. If the wave function is real, this essentially is psi squared.

Note that the probability density function is squared because the wave function includes the complex number, i or sqrt(-1).

The probability for finding the particle between x + dx is proportional to

The probability for finding the particle between x₁ and x₂ is the integral from x₁ to x₂ of |ψ(x)|²dx.

The probability for finding the particle anywhere is the integral from -∞ to ∞ of |ψ(x)|²dx.

We set the integral equal to one because if the particle exists, the chance of it being somewhere is 100%, or 1/1 in fraction form. This corresponds to one. We normalize the wave function by adjusting constants so that the integral described above equals one.

Fig 1: Graph of Probability Density Function vs. Position, x.

The Uncertainty or Indeterminacy Principle

A particle's momentum and position can't be measured at the same time; this is Heisenberg's uncertainty principle. We can know the exact value of position or the momentum but the other value is indeterminate.

Why are we learning about this in a Chem class?

The wave functions for electrons in an atom are called atomic orbitals. Understanding the properties of atomic orbitals is crucial to understanding chemical bonding.

Properties of the Wave Function

The properties of the wave function include: it only has one value at any point, it is continuous, it is finite at any point, it has a continuous first derivative and Schrodinger requires that the second derivative is well-defined as well. Wave functions have nodes where the values cross the x-axis, or where it equals zero. The probability density function has the same nodes as the wave function.

Fig. 2: A graph of a wave function, followed by the wave function squared. As you can see, the wave function and the wave function squared have the same nodes.

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Time Independent Wave Functions:

Classical mechanics relates the second derivatives of waves with respect to distance to second derivatives with respect to time. It is much simpler to only consider the wave function with respect to distance, or time independent wave functions.

In this course, we focus on time independent wave functions. Using an example of a wave function, we can justify this.

Examples of time independent wave functions:

As a reminder, to normalize the wave function, we adjust the value of A or B so that the integral of the function squared equals one.

The derivative of the sine function is

Which we obtain using our knowledge that the derivative of sine is cosine and then using the chain rule, multiplying the derivative of the outer function by the derivative of the inner function.

We can then take the derivative of the derivative to find the second derivative.

Using the chain rule as before and that the derivative of A cos(f(x)) is –A sin(f(x)).

Note that includes the original wave function,

The second derivative of the wave function is the wave function multiplied by a constant.

This is also true if we start with

This fact allows us to only consider time independent wave functions for the purpose of this class, as the second derivative of the wave function can be related to the wave function itself.

Uncertainty and Expectation Values

Since the wave function mainly deals with probabilities, the quantities related to the wave function are therefore statistical in nature.

"The expectation value of position," which is the average value over the probability distribution, is described mathematically by:

If you review basic Calculus, you should recognize this as the equation for the average value of a function.

Since we like to talk about squared values when dealing with wave functions, "the expectation value of position squared" can also be calculated.

Not that this is the same equation except that x is not squared in the integral.

Since statistics is involved, we have to mention standard deviation, called "uncertainty." The standard deviation of position from the average value is called "the position uncertainty."

Note that the values involved in this equation are x and the average of x.

Fig 3: Graph of Probability Density Function vs. x Displaying the Expectation Value of Position and the Position Uncertainty.

The Schrödinger Equation

Fig. 3: The fundamental equation of Wave Mechanics, the Schrodinger equation in one dimension, which relates the wave function to energy.

The Schrödinger equation is essentially the Hamiltonian operator on the quantum level and is similar to Newton’s Second Law.

If we have the initial condition of a particle of mass m moving along the x-axis and acted upon by a force F(x,t), we can use Newton’s second law (F=ma) to determine the particle’s position, velocity, and kinetic energy. Similarly, we can use the Schrödinger equation to predict the state of the particle at any other time. The equation gives the relationship between the wave function and energy.

We begin with two known equations

wavelength is equal to Planck’s constant over momentum, from the de Broglie equation.