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Quantum - infinite chain of wells

  1. Jul 23, 2017 #1
    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. jcsd
  3. Jul 24, 2017 #2

    Orodruin

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    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
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