Superposition

A photon can have a probability of x%x\% of spinning up (+). This means it would have (100−x)%(100-x)\% of spinning down (-), since the probabilities must add up. The photon actually is both spinning up and spinning down until we observe it, where it collapses to either spinning up or spinning down. Schrödinger’s cat.

We call this position of spinning both up and down simultaneously a “superposition.”

Lets just say the kets ∣+⟩\left\lvert + \right\rangle is the state of spinning up and ∣−⟩\left\lvert - \right\rangle is the state of spinning down.

Let a “state” just be a N×1N\times 1 matrix like what we just said above. So ∣x⟩,∣ψ⟩,∣ϕ⟩\left\lvert x \right\rangle,\left\lvert \psi \right\rangle,\left\lvert \phi \right\rangle are all states.

Some states can be made of other states.

Example:

Let

∣x⟩=(10)∣y⟩=(01)\begin{gather*} \left\lvert x \right\rangle=\begin{pmatrix} 1 \\ 0 \end{pmatrix}\quad \left\lvert y \right\rangle=\begin{pmatrix} 0 \\ 1 \end{pmatrix} \end{gather*}

If i multiply a matrix by a number (let this be called a “scalar”), we apply the multiplication across all values.

⇒3∣x⟩=3(10)=(3×13×0)=(30)\begin{gather*} \Rightarrow 3\left\lvert x \right\rangle=3\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}3\times1\\3\times0\end{pmatrix}=\begin{pmatrix}3\\ 0\end{pmatrix} \end{gather*}

If we add these matrices together ∣x⟩+3∣y⟩\left\lvert x \right\rangle+3\left\lvert y \right\rangle we get

⇒∣x⟩+3∣y⟩=(10)+3(01)=(13)\begin{gather*} \Rightarrow\left\lvert x \right\rangle+3\left\lvert y \right\rangle=\begin{pmatrix} 1 \\ 0 \end{pmatrix}+3 \begin{pmatrix} 0 \\ 1 \end{pmatrix} =\begin{pmatrix} 1 \\ 3 \end{pmatrix} \end{gather*}

Notice how we’re adding the nn-th elements of the states together for each row?

We can represent these two states coexisting (i.e., a superposition) using math now!

∣ψ⟩=α∣+⟩+β∣−⟩\begin{gather*} \left\lvert \psi \right\rangle=\alpha\left\lvert + \right\rangle+\beta\left\lvert - \right\rangle \end{gather*}

where α,β\alpha,\beta are probability amplitudes. See how they are a combination of up and down states?

There is ∣α∣2|\alpha|^2 probabilty that the photon, once observed, spins upwards and ∣β∣2|\beta|^2 probability it spins downwards once observed.

∣a∣|a| is just a way of saying make aa positive if it is negative.

Example:

∣(−3)∣=3∣5∣=5\begin{gather*} |(-3)|=3\\ |5|=5 \end{gather*}

Note that

∣α∣2+∣β∣2=1\begin{gather*} |\alpha|^2+|\beta|^2=1 \end{gather*}

as you can’t have more than 100% or less than 0% probability of something existing.

I’m going to write a ton of mathematical definitions which will build up how to do the physics in the next section!