Quantum Mechanics

This is a topic from Higher Physics 1B

Introduction

Quantum mechanics is the branch of physics dealing with physical phenomena at microscopic scales, particularly with notions of probability.

Probability in Matter

• Considering light as particles (photons), the probability per volume of finding a photon in a given region of space at a given time is proportional to the number N of photons per unit volume at that time and to the intensity:

• Considering light as a wave, the intensity is proportional to the magnitude of the electric field ( I α E²)
• Combining these perspectives gives:

• This equation means that the probability per unit volume of finding a photon in a given region is proportional to the square of the amplitude of the corresponding EM wave. This amplitude is called the probability amplitude or wave function and denoted Ψ

Wave Function

The complete wave function for a system is dependant upon the positions of all the particles which make up that system, for example the function for the j_th particle within a system of t particles is given as:

Ψ(r₁, r₂ ... r_j ... r_t) = Ψ(r_j)e^-iωt

Where r_j is the position of the j_th particle in the system

ω = 2πf is the angular frequency

i = -1^0.5

t is the total number of particles in the system

The Absolute Square of the Wave Function

• The wave function is often of complex value, and so we consider the absolute square (|Ψ|² = Ψ*Ψ where Ψ* is the complex conjugate) instead which is always real and positive
• |Ψ|² is proportional to the probability per unit volume of finding a particle in a given region at a given time - in general the probability of finding a particle in small incremental volume dV is |Ψ|²dV
• In one dimension this becomes |Ψ|²dx such that the probability of finding a particle in the interval a ≤ x ≤ b is

which equals the area under the curve

• The wave function for a free particle moving along the x-axis is given as:

Ψ(x) = Ae^ikx

Where A is a constant amplitude

k = 2π/λ is the angular wave number

Normalisation

• |Ψ|² is a probability density function for continuous discrete variables (see MATH1231 Probability and Statistics Course Notes) such that:

• A wave function which satisfies this equation is said to be normalised, the implication of which is that it exists at some point in space

Expectation Values

• Ψ is not used as a measurable quantity in and of itself, but rather it is used to derive other measurable quantities, such as the expectation value of x, which is its average position and is defined as follows:

• The expectation value for any function of x is given similarly as:

Summary

• Ψ(x) can be complex or real
• Ψ(x) must be defined and single-valued at all points in space
• Ψ(x) must be normalised
• Ψ(x) must be continuous

The 'Particle in a Box' Thought Experiment

Consider a particle confined to a one-dimensional region of space, by a one-dimensional 'box' such that it is bouncing back and forth between two impenetrable walls L metres apart.

Deductions:

• As long as the particle remains inside the box, the potential energy is independent of location and can be set to zero
• It is impossible for the particle to exist outside the box (Ψ(x) = 0 for x<0 and x>L), the implication of which is that if the particle was to be found outside

of the box it would have infinite energy

• The wave function Ψ(x) is always continuous, and so if Ψ(x) = 0 for x<0 and x>L, then Ψ(0) = 0 and Ψ(L) = 0

Conclusions:

• The wave function can be expressed as a real sinusoidal function:

Ψ(x) = A sin(2πx / λ) = A sin(nπx / L)

• The wavelengths and corresponding absolute square wave functions are quantised, as depicted in the following figure:

Energy for the 'Particle in a Box'

• Potential energy inside the box was set at zero, so all the particle's energy is kinetic energy and is quantised, given as:

Schrödinger's Equation

The physicist Schrödinger applied De Broglie's equation to the probability wave equation to create an equation which describes the probability for a particle or wave in one dimension. The equation is complicated, as is its application, however the derivation is quite reasonable and will help in the understanding of quantum mechanics and in answering any questions pertaining to the equation in exam papers with confidence. The time-independent Schrödinger equation is as follows:

Where ℏ = h / 2π

m represents mass

Ψ(x) is the wave function for a particle in one dimension

x is position in one dimension

U(x) is some function for the potential energy of the particle at position x

E is the total energy of the particle (both kinetic and potential)

Derivation

• Begin by recalling the equation for the wave function in one dimension, and deriving it in terms of x, twice:

• The second derivative can be rewritten in terms of the original function:

• Given that all waves are also particles, we can substitute De Broglie's equation:

• Recalling that total energy for a particle equals kinetic energy plus potential energy (E = K + U), and recalling that K = mv²/2 = (mv)²/2m = p² / 2m:

E = p² / 2m + U(x) (expressing potential energy as a function of position)

p² = 2m[E - U(x)]

• Substituting p² into what we have derived so far and rearranging gives the equation:

Schrödinger's Equation and the 'Particle in a Box'

• In the region 0 ≤ x ≤ L, where U = 0, the equation is simplified to:

• The solution to this, (as seen in the first part of the derivation) is a wave equation:

Ψ(x) = A sin(kx) + B cos(kx)

Where the constants A and B are determined by the boundary and normalisation conditions

• A consideration of Ψ(x) for different allowed energy levels gives:

Finite Potential Wells

A finite potential well is a concept much like the 'particle in a box'.

• The energy is zero in region II
• The energy has a finite value outside of the well (regions I and III)
• The general solution is:

Ψ(x) = Ae^Cx + Be^-Cx

Where A, B and C are constants

• In region I, B = 0 necessarily in order to avoid infinite energy for large negative values of x
• In region III, A = 0 necessarily in order to avoid infinite energy for large positive values of x
• A practical example of a finite energy well that may be referred to in exam questions is the quantum dot, a region used in nanotechnology which acts like a finite energy well
• Classical physics suggests that it is impossible for the particle to somehow pass through the barriers of the wall instead of being reflected, but sometimes this happens. This process is called tunnelling or barrier penetration and it is seen in Alpha decay and in nuclear fusion
• The probability of tunnelling is given by the transmission coefficient T, while the probability of reflection is given by the reflection coefficient R, such that:

T + R = 1

Quantum Numbers

Electron orbital states can be defined by a set of quantised values known as quantum numbers, of the form (n, l, m) where:

n is the principal/radial number (the energy level)

l is the angular number

m_l is the projection/magnetic moment