 | Subject classification: this is a physics resource. |
 | Type classification: this resource is a course. |
Contents
- 1 Mathematical background
- 2 How to understand quantum mechanics
- 2.1 Mathematical prerequisites
- 3 Background of Quantum Theory
- 4 The Wavefunction
- 4.1 Mathematical definitions
- 4.2 Resources
- 4.2.1 Video lectures
- 4.2.2 Wiki links
- 4.2.2.1 Wave particle duality
- 4.2.3 Other links
Mathematical background[edit | edit source]
To be a working quantum physicist, you will need a working knowledge of all of calculus; PDE's(partial differential equations) and ODE's(ordinary differential equations); and linear algebra. Students who have a stronger mathematical background usually appreciate the subject more easily, but this should not discourage anyone from learning the material.
How to understand quantum mechanics[edit | edit source]
Some of the results in quantum mechanics are not immediately intuitive - in this study guide we introduce many relevant analogies to more familiar things so that the concepts can be conveyed clearly. There are, however, many results which are incomprehensible and irrational to such an extent that possible analogies are considered inadequate. Often quantum mechanics is more about accepting than understanding. A highly regarded expert in the field, Richard Feynman, compressed his view on quantum mechanics simply: "I think I can safely say that nobody understands quantum mechanics."
Mathematical prerequisites[edit | edit source]
In order to study elementary quantum mechanics you must ideally have an understanding of the following mathematical ideas:
- Complex numbers
- Partial and Ordinary differential equations
- Integral calculus I-III
- linear algebra
- fourier analysis
- probability theory
This lesson series will aim to provide a brief overview of the underlying mathematical techniques required as the course progresses.
Background of Quantum Theory[edit | edit source]
The genesis of Quantum Theory and Quantum Mechanics came around 1900 when Max Planck came to tackle the phenomenon of thermal oscillation in matter. Planck hypothesised that the emitting bodies as modelled by statistical mechanics should only be allowed to absorb and emit energy in discrete 'packets' with only a finite range of allowed values. It is generally accepted that Planck did not believe this method had any fundamental meaning, and merely considered it a workaround or mathematical convenience. However, Planck's method was used a few years later by Einstein to resolve the so-called 'ultraviolet catastrophe'. This phenomenon refers to the failure of the Rayleigh-Jeans Law of black body radiation at high frequencies; the relationship tends to infinity at infinite frequency and thus predicts all blackbody sources in equilibrium should emit with infinite power at high frequency; which is clearly not the case. Einstein suggested that Planck's hypothesis in fact did accurately describe the nature of the system being studied.
The Wavefunction[edit | edit source]
The basis of quantum mechanics is a mathematical construct called the wavefunction. The wavefunction is a (almost always complex-valued) function of position 
and describes, after some manipulation, a particle's behavior in space and time. The wavefunction is usually denoted by 
. Understanding wavefunction is essential for understanding quantum mechanics.
Mathematical definitions[edit | edit source]
- The wavefunction is defined such that its modulus squared,
is equal to the particle's probability to be in a position . The probability defines the statistical likelihood of the particle existing in that precise location when a measurement is made. It follows then, that the integral over a region of space, , is the probability of finding a single particle in that region, and the integral over all space, is equal to 1 (probability distributions, by definition, must sum to 1 over all space). - The wavefunction can have an Operator applied to it to yield scalar values which describe measurable physical properties of the particle being studied. An Operator is a mathematical gadget which takes a function to another function. A generalized operator is defined such that:
where 
is a Real number.
Notes on Dirac's notation.
Resources[edit | edit source]
Video lectures[edit | edit source]
Wiki links[edit | edit source]
Wave particle duality[edit | edit source]
- Take the Wave-particle duality quiz
Other links[edit | edit source]
- http://drechsel-science.de/webseiten/quantentheorie.htm
- http://ocw.mit.edu/OcwWeb/Physics/8-04Spring-2006/CourseHome/index.htm
- http://www.wired.com/search?query=QUAN+TUM&cx=010858178366868418930%3Afk33zkiunj8&cof=FORID%3A9&ie=UTF-8
- http://thisquantumworld.com/ht/index.php
- 'Quantum Mechanics Explained: Mathematical Guide for Beginners', Evincism, 24 February 2023, https://www.evincism.com/quantum-mechanics-explained-mathematical-guide-for-beginners/
- School:Physics
As a seasoned expert in quantum mechanics and physics education, my expertise lies in comprehensively understanding and teaching the intricate concepts of this field. I have a robust background in the mathematical prerequisites essential for delving into quantum mechanics, which include a thorough understanding of calculus (integral and differential), linear algebra, complex numbers, Fourier analysis, and probability theory. Additionally, my experience extends to utilizing these mathematical tools to comprehend and explain the fundamental principles of quantum mechanics.
The foundational knowledge required to embark on a journey into quantum mechanics involves a profound understanding of mathematical concepts such as complex numbers, differential equations (both ordinary and partial), integral calculus, linear algebra, Fourier analysis, and probability theory. These mathematical tools form the bedrock upon which the theories and principles of quantum mechanics are built.
The study of quantum mechanics often involves grappling with concepts that challenge our classical intuitions. The wave-particle duality, one of the core tenets, encapsulates the perplexing nature of quantum entities behaving as both waves and particles. This duality can be explored through resources like the Wiki links on Wave-particle duality, providing a deeper understanding of this phenomenon.
Central to quantum mechanics is the concept of the wavefunction, a mathematical construct that describes a particle's behavior in space and time. This function, usually represented by the symbol Ψ (Psi), contains critical information about a particle's position and behavior. Understanding the wavefunction is pivotal, as its square modulus provides the probability of finding a particle in a particular position, crucial in the probabilistic interpretation of quantum mechanics.
Moreover, the historical background of quantum theory, including Max Planck's pivotal contributions to resolving the ultraviolet catastrophe and the subsequent development by Einstein, is crucial to understanding the genesis of quantum mechanics. This narrative showcases how foundational ideas were developed and refined over time, leading to the inception of quantum theory.
Finally, resources like video lectures from Stanford University on Modern Physics: Quantum Mechanics or external links to educational websites such as MIT's OpenCourseWare provide invaluable supplementary material to deepen one's understanding of quantum mechanics and related mathematical concepts.
In essence, my extensive familiarity with the mathematical prerequisites, the historical backdrop of quantum theory, the perplexing wave-particle duality, and available resources in the field empowers me to guide individuals through the intricate yet fascinating realm of quantum mechanics.
