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I Real world process that results in reciprocal wavefunction

  1. Mar 31, 2017 #1
    Two Questions from a newbie.

    A) Is there a easily implemented process or reaction that results in a particle with reciprocal wave function of input particle?
    B) Is there a easily implemented process or reaction that results in a particle A transferring it's wavenumber and angular frequency to particle B?

    Any help appreciated.
     
    Last edited: Mar 31, 2017
  2. jcsd
  3. Apr 1, 2017 #2

    blue_leaf77

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    Homework Helper

    Do you mean reciprocal as in ##1/x## being the reciprocal of ##x##? In that case, I really suspect the output wavefunction will be normalizable and hence it can be an indication that such process might not exist.
     
  4. Apr 1, 2017 #3
    I can think of one. Have absolutely no idea if it is meaningful.

    if you form the matrix
    ##\def \sqx{\frac{1}{\sqrt{2}}}##
    ##
    \begin{bmatrix}
    \frac{1}{\psi_1^2} & 0 & 0 & 0 & 0 & 0\\
    0 & \frac{1}{\psi_2^2} & 0 & 0 & 0 & 0\\
    0 & 0 & \frac{1}{\psi_3^2} & 0 & 0 & 0\\
    0 & 0 & 0 & \frac{1}{\psi_4^2} & 0 & 0\\
    0 & 0 & 0 & 0 &\frac{1}{\psi_5^2} & 0\\
    0 & 0 & 0 & 0 & 0 & \frac{1}{\psi_6^2}
    \end{bmatrix}\begin{bmatrix}
    \psi_1\\
    \psi_2\\
    \psi_3\\
    \psi_4\\
    \psi_5\\
    \psi_6
    \end{bmatrix} = \begin{bmatrix}
    \\\frac{1}{\psi_1}\\
    \\\frac{1}{\psi_2}\\
    \\\frac{1}{\psi_3}\\
    \\\frac{1}{\psi_4}\\
    \\\frac{1}{\psi_5}\\
    \\\frac{1}{\psi_6}
    \end{bmatrix}##
     
  5. Apr 1, 2017 #4

    Strilanc

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    A)

    If you mean some operation ##F## such that ##F(\psi) = 1/\psi##, then no. The operation doesn't preserve total squared amplitude. It isn't unitary.

    If you mean a conjugation operation ##F## such that ##F(\psi) = \psi^*##, then closer but still no. The operation does preserve total squared amplitude, but it's not linear w.r.t. the amplitudes. It can separate the real and imaginary parts of the amplitudes from each other. It's not unitary. Actually, given this kind of operation, you can build an FTL communication mechanism.

    B)

    I don't know about "easily implemented", but in quantum computing we have the SWAP gate. It swaps the values of two qubits. Physically, that could mean just literally moving a particle to where another was and vice versa, or some more complicated protocol involving a teleportating through a few CNOT gates, or lots of other things. Quantum information is generally pretty fungible.
     
  6. Apr 1, 2017 #5
    Thanks for replies, so no operation exists to negate both wavenumber and angular frequency? (resulting in reciprocal). Could one still apply 2 operations, one for each, because surely "direction" and "rotation" could be changed one at a time to achieve the goal?
     
  7. Apr 1, 2017 #6

    Strilanc

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    Science Advisor

    Oh, you're talking about permuting the basis states not operation on the individual amplitudes.
     
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