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Rewrite state in new basis - Quantum Mechanics

  1. May 10, 2015 #1
    1. The problem statement, all variables and given/known data
    Rewrite the state |ψ⟩ = √(1/2)(|0> + |1>) in the new basis.

    |3⟩ = √(1/3)|0⟩ + √(2/3)|1⟩

    |4⟩ = √(2/3)|0⟩ − √(1/3)|1⟩


    You may assume that |0⟩ and |1⟩ are orthonormal.

    2. Relevant equations

    3. The attempt at a solution

    I have a similar example in my notes however there is a step that I has stumped me. Annoyingly its the first one.

    In my notes I have:

    """If we want to work in the basis |+⟩ and |−⟩ instead of | ↑⟩ and |↓⟩, with,

    |+⟩ = (1/√2)(| ↑⟩ + | ↓⟩) & |−⟩ = (1/√ 2)(| ↑⟩ − | ↓⟩)

    how would |ψ⟩ and I be written in the new basis?

    Let us rearrange as:

    | ↑⟩ = 1/(√2)(|+⟩ + |−⟩) & | ↓⟩ = (1/√2)(|+⟩ − |−⟩)"""

    After rearranging I think that I should be able to complete the question but as it stands I cant see how to rearrange them to get |0> & |1>. Any advice would be much appreciated as really struggling with Dirac notation at the moment. Thanks 12x4
     
  2. jcsd
  3. May 10, 2015 #2

    vela

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    You're probably just getting confused by the new notation. Consider the ordinary algebraic equations
    \begin{align*}
    u &= \frac{1}{\sqrt 2} x + \frac{1}{\sqrt 2} y \\
    v &= \frac{1}{\sqrt 2} x - \frac{1}{\sqrt 2} y
    \end{align*} How would you solve for ##x## in terms of ##u## and ##v##? You can essentially do the same thing.
     
  4. May 10, 2015 #3
    thanks vela, I just managed to do it with your advice.
     
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