Basic Principles of Quantum Mechanics

Table of Contents:

  1. Introduction
  2. Rules of Constructing Operators
  3. Commuting/Noncommuting Operators

INTRODUCTION

One of the most basic principles of quantum mechanics is that the state of the system can be completely described by a state vector, or function, normally denoted as $\psi$. The state function or vector is analogous to the equation for a plane wave, in that they both provide the solution to a wave function. In Quantum Mechanics, the wave function is known as the Schrodinger equation. We will get to that in a bit, though, as there are a few more principles that we need to learn first in order to fully appreciate the Schrodinger equation!

The wave function $\psi$ is a vector belonging to Hilbert space, which means that the wave function possesses all of the properties of Hilbert space in terms of completeness and inner product rules. The understanding of the wave function that is required to understand the quantum mechanical aspects of lasers does not require a complete knowledge of Hilbert space, but Wikipedia has a good article with references that could be useful here. We will just be using the resulting properties of Hilbert space functions for our purposes here.

The wave function may be multiplied by an arbitrary complex constant and still represent the same physical state. Normalization is normally required to provide an orthonormal set of bases, but this just changes the amplitude of the wave function while leaving its phase arbitrary.

The superposition principle for the wave function then states that the sum of two state vectors is another state vector belonging to the same Hilbert space. The sum of the two state vectors $c_1\psi_1 + c_2\psi_2$ means that the system is simultaneously in both states, with the probability of measuring either state upon collapsing the wave function given by the normalized amplitudes $c_1^2+c_2^2 =1$. This would be true for any number of superposition of states.

This then encapsulates some of the main differences between quantum and classical physics. There is a nonlocal nature of quantum mechanics except when measuring. There are other paradoxes such as EPR that also highlight the fundamentally non-local nature of Quantum Mechanics.

An important note is that although the absolute phase of a state vector $\phi$ is not physically relevant, the relative phase between terms is relevant because it encodes some important physical information.

Another one of the most important principles of quantum mechanics is that for every observable physical quantity there corresponds a linear Hermitian operator. Any measurement of an observable, physical quantity such as momentum, position, energy, angular momentum, etc., must then result in a number that belongs to the eigenvalue set $q_i$ of the corresponding operator $\hat{Q}$.

Mathematically, we can find all possible results of a measurement by solving the eigenvalue equation:

\begin{equation} \hat{Q}\cdot\psi_i = q_i\psi_i \end{equation}

The eigenvalue $q_i$ is therefore associated with the eigenfunction $\psi_i$, exactly as in linear algebra. Because all of the observable physical quantities are represented by a real eigenvalue $q_i$, the corresponding operators $\hat{Q}$ must be Hermitian. As such, the Hermitian operators constitute a complete set of orthogonal basis vectors into which any arbitrary state vector $\psi$ can be decomposed. If we enforce state vector normalization:

\begin{equation} \int\psi^*(r,t)\psi(r,t)d^3r = 1 \end{equation}

Then we have the conditions that:

  1. If $\psi(r,t) = \psi_i(r,t)$, where $\psi_i(r,t)$ is an eigenfunction of $\hat{Q}$, a measurement of the observable Q is certain to yield the corresponding eigenvalue $q_i$.
  2. If $\psi(r,t) = c_1\psi_1+c_2\psi_2+ … + c_n\psi_n$, where each $\psi_i(r,t)$ is an eigenfunction of $\hat{A}$, the probability that a measurement of A will yield the value $a_i$ is:
    \begin{equation} P(a_i) = |c_i|^2 = |\int\psi^*_i(r,t)\cdot\psi(r,t)d^3r|^2 \end{equation}

Due to the enforced normalization, $\sum^n_{i=1}|c_i|^2=1$. The average value of a measurement of a system when it is the state $\psi$ is then given as:

\begin{equation} <A> = \sum_{i=1}^nP(a_i)a_i = \sum_{i=1}^{n}|c_i|^2a_i = \int\psi^*(r,t)\hat{A}\psi(r,t)d^3r \end{equation}

RULES OF CONSTRUCTING QUANTUM MECHANICAL OPERATORS

Since any classical physical quantity Q can be expressed as a function of position and momentum Q = Q(x,p), we can replace x with the operator $\hat{x}$ and p with the operator $\hat{p}$ in the classical expression such that we have $\hat{Q} = Q(\hat{x},\hat{p})$. The basic operators for $\hat{x}$ and $\hat{p}$ can then be defined as:

\begin{equation} \hat{x}\psi = x\psi \end{equation}

\begin{equation} \hat{p}\psi = -i\hbar\frac{\partial\psi}{\partial{x}} \end{equation}

It should be noted that there are also purely quantum mechanical operators, such as the spin operator, than cannot be obtained through substitution of $\hat{x}$ and $\hat{p}$. Each of these needs to be “guessed” individually depending on the specific quantity of measurement.

Since in classical mechanics we could use the Hamiltonian to describe the amount of energy in a system, we can do the same for quantum mechanics with our substitution of the canonical position and momentum operators. For a conservative system, the Hamiltonian is given as:

\begin{equation} \hat{H} = \frac{1}{2m}\hat{p}^2+V(\hat{x}) \end{equation}

where “m” is the particle mass and V is the potential.

COMMUTING AND NONCOMMUTING OPERATORS

With the fact that a measurement of a system yields one specific eigenvalue of an operator, it is then an important aspect of quantum mechanics that there are certain measurements of a system that would “interfere” with one another. We can describe this using a commutation relation as:

\begin{equation} [\hat{A},\hat{B}] = \hat{A}\hat{B} – \hat{B}\hat{A} \neq 0 \end{equation}

Consequently, this means that the quantities A and B cannot be measured with arbitrary (e.g. exact) accuracy simultaneously. One important example of this is position and momentum. By directly substituting the position and momentum operators we can calculate:

\begin{equation} [\hat{x},\hat{p}]\psi(x,t) = -i\hbar(x\frac{\partial\psi}{\partial{x}}-\frac{\partial}{\partial{x}}(x\cdot\psi)) = i\hbar\psi(x,t) \end{equation}

Some examples of operators that would commute are position and spin, or momentum and spin. It can be shown that commuting operators share common eigenfunctions, and operators with common eigenfunctions commute.

This all brings us, finally, to the Schrodinger equation! The Schrodinger equation is given as:

\begin{equation} i\hbar\frac{\partial}{\partial{t}}\psi(r,t) = \hat{H}\psi(r,t) \end{equation}

The Schrodinger equation governs the time evolution of the system. If the Hamiltonian for a certain system is time independent, then we may write the time-independent Schrodinger equation (analogous to the Helmholtz equation) as:

\begin{equation} \hat{H}\psi_i(\vec{r}) = E_i\psi_i(\vec{r}) \end{equation}

This means that we can write the state vector as a separable function of position and time as:

\begin{equation} \psi_i(r,t) = \psi_i(\vec{r})e^{-\frac{i}{\hbar}E_it} \end{equation}

The general expression for a state-vector of such a system is:

\begin{equation} \psi(r,t) = \sum_ic_i\psi_i(\vec{r})e^{-\frac{i}{\hbar}E_it} \end{equation}

The probability of finding the system in the state $\psi_r(\vec{r})$ with energy $E_i$ is $P_i = |c_i|^2$. The relation $[\hat{Q},\hat{H}] = 0$ means conservation of the physical property Q. If the system is initially in an eigenstate of operator $\hat{Q}$ it remains so during its time evolution.

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