1.3: The Superposition Principle
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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⟩, and the succesive excited states by |1⟩,…,|k−1⟩. 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 α0|0⟩+α1|1⟩+⋯+ αk−1|k−1⟩, where α0,α1,…,αk−1 are complex numbers normalized so that ∑j|αj|2=1.αj is called the amplitude of the state |j⟩. For instance, if k=3, the state of the electron could be
|ψ⟩=1√2|0⟩+12|1⟩+12|2⟩
or
|ψ⟩=1√2|0⟩−12|1⟩+i2|2⟩
or
|ψ⟩=1+i3|0⟩−1−i3|1⟩+1+2i3|2⟩.
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 α0 is a measure of its inclination towards the ground state. Of course we cannot think of α0 as the probability that an electron is in the ground state - remember that α0 can be negative or imaginary. The measurement priniciple, which we will see shortly, will make this interpretation of α0 more precise.