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Electron in two Potential Wells

  1. May 31, 2014 #1
    1. The problem statement, all variables and given/known data

    An electron can be in one of two potential wells that are so close that it can “tunnel” from one to the other (see §5.2 for a description of quantum- mechanical tunnelling). Its state vector can be written
    |ψ⟩ = a|A⟩ + b|B⟩, (1.45)
    where |A⟩ is the state of being in the first well and |B⟩ is the state of being in the second well and all kets are correctly normalised. What is the probability of finding the particle in the first well given that: (a) a = i/2; (b) b = e^(i*pi); (c) b = 1/3 + i/√2?

    2. Relevant equations

    a*a is the probability of finding a particle in state A

    3. The attempt at a solution

    The question is confusing me. I don't know what the second b is for. Also, these are supposed to be normalized according to the question, but b*b (for the first b) would be 1 all by itself. Is this question ok, and I am just missing something?

    Thanks,
    Chris Maness
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 31, 2014 #2

    hilbert2

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    Actually, the probability to be in state ##A## is ##|a|^{2}## (square modulus, different from simple square for complex numbers). If ##|b|## has value ##1##, it just means that ##a## must be zero if the state is normalized. If one of the constants ##a,b## is given, you can deduce the absolute value of the another from the normalization condition.
     
  4. May 31, 2014 #3
    Yes, I show that the probability for finding the system in state a is a*a. Where a* is the complex conjugate of a. This is the mod squared for a complex number. The only problem I am having is that he states that it is already normalized. I expect |a|^2 +|b|^2=1, but it is not, and then there is another b too. This confuses me more.

    Chris
     
  5. May 31, 2014 #4

    hilbert2

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    Sorry, I confused the star "*" with a multiplication sign. The parts (a), (b) and (c) are separate problems and have different states of form ##a\left|A \right> + b\left|B\right> ## as answers. You are not given two values of ##b## for solving the same problem.
     
  6. May 31, 2014 #5
    Perfect, thanks. I have been doing too many math problems that have the phrase "show that" in them :D

    Chris KQ6UP
     
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