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Physics LibreTexts

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 k1 excited states. As a classical system, we might use the state of this system to store a number between 0 and k1. 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,,|k1. 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++ αk1|k1, where α0,α1,,αk1 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=12|0+12|1+12|2

or

|ψ=12|012|1+i2|2

or

|ψ=1+i3|01i3|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 k1 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.


1.3: The Superposition Principle is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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