1.3: The Superposition Principle

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

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

or

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

or

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

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.