# Quantum - infinite chain of wells

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1. Jul 23, 2017

### Toby_phys

An electron can tunnel between potential wells. Its state can be written as:

$$|\psi\rangle=\sum^\infty_{-\infty}a_n|n\rangle$$

Where $|n \rangle$ is the state at which it is in the $n$th potential well, n increases from left to right.

$$a_n=\frac{1}{\sqrt{2}}\left(\frac{-i}{3}\right)^{|n|/2}e^{in\pi}$$

What is the probability of finding the election in well $0$ or above?

Is this probability
$$|a_0|^2+|a_1|^2+...$$
or is it $$|a_0+a_2+...|^2$$?

I am leaning towards the first option but this doesn't use the exponential phase factor.

My reasoning is, let $$|\phi \rangle$$ be the superposition of everything 0 and above:
$$|\phi\rangle=\frac{\sum^{\infty}_{0}a_n|n\rangle}{\sqrt{\sum^{\infty}_0|a_n|^2}}$$

The denominator normalises everything.
Taking the inner product we get:

$$\langle\phi|\psi \rangle=\frac{\sum^{\infty}_{0}|a_n|^2}{\sqrt{\sum^{\infty}_0|a_n|^2}}=\sqrt{\sum^{\infty}_0|a_n|^2}$$

The mod square of this is the sum of the individual probabilities.

2. Jul 24, 2017

### Orodruin

Staff Emeritus
This one.

The phase factor will only be relevant if you start evolving the state with a Hamiltonian that is not diagonal in the given basis or want to know if the system is in a particular state that is a different linear combination.

Last edited: Jul 24, 2017