1.3: The Superposition Principle

Last updated
Save as PDF

Page ID: 92688

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Consider a system with \(k\) distinguishable (classical) states. For example, the electron in a hydrogen atom is only allowed to be in one of a discrete set of energy levels, starting with the ground state, the first excited state, the second excited state, and so on. If we assume a suitable upper bound on the total
energy, then the electron is restricted to being in one of \(k\) different energy levels - the ground state or one of \(k-1\) excited states. As a classical system, we might use the state of this system to store a number between 0 and \(k-1\). The superposition principle says that if a quantum system can be in one of two states then it can also be placed in a linear superposition of these states with complex coefficients.

Let us introduce some notation. We denote the ground state of our \(k\)-state system by \(|0\rangle\), and the succesive excited states by \(|1\rangle, \ldots,|k-1\rangle\). These are the \(k\) possible classical states of the electron. The superposition principle tells us that, in general, the quantum state of the electron is \(\alpha_{0}|0\rangle+\alpha_{1}|1\rangle+\cdots+\) \(\alpha_{k-1}|k-1\rangle\), where \(\alpha_{0}, \alpha_{1}, \ldots, \alpha_{k-1}\) are complex numbers normalized so that \(\sum_{j}\left|\alpha_{j}\right|^{2}=1 . \alpha_{j}\) is called the amplitude of the state \(|j\rangle\). For instance, if \(k=3\), the state of the electron could be

\[|\psi\rangle=\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{2}|1\rangle+\frac{1}{2}|2\rangle\]

\[|\psi\rangle=\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{2}|1\rangle+\frac{i}{2}|2\rangle\]

\[|\psi\rangle=\frac{1+i}{3}|0\rangle-\frac{1-i}{3}|1\rangle+\frac{1+2 i}{3}|2\rangle .\]

The superposition principle is one of the most mysterious aspects about quantum physics - it flies in the face of our intuitions about the physical world. One way to think about a superposition is that the electron does not make up its mind about whether it is in the ground state or each of the \(k-1\) excited states, and the amplitude \(\alpha_{0}\) is a measure of its inclination towards the ground state. Of course we cannot think of \(\alpha_{0}\) as the probability that an electron is in the ground state - remember that \(\alpha_{0}\) can be negative or imaginary. The measurement priniciple, which we will see shortly, will make this interpretation of \(\alpha_{0}\) more precise.