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We know that superposition principle is valid in Quantum Mechanics

  1. Jul 21, 2013 #1
    because Schrodinger equation is a linear differential equation. How do we show that Schrodinger equation is a linear differential equation?
     
  2. jcsd
  3. Jul 21, 2013 #2

    hilbert2

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    If you have a second order PDE for a function [itex]\psi(x,t)[/itex], it is of the general form

    [itex]F(\partial_{tt}\psi,\partial_{xx}\psi,\partial_{xt}\psi,\partial_{t} \psi,\partial_{x}\psi,\psi)=g(x,t)[/itex]

    If the equation is linear, then [itex]F[/itex] is linear with respect to all its arguments, i.e.

    [itex]F(\partial_{tt}\psi,\partial_{xx}\psi,\partial_{xt}\psi,\partial_{t} \psi,\partial_{x}\psi,\psi,x,t)\\=a(x,t)\partial_{tt}\psi+b(x,t) \partial_{xx} \psi+c(x,t)\partial_{xt}\psi+d(x,t)\partial_{t} \psi+e(x,t)\partial_{x}\psi+f(x,t)\psi[/itex]

    There's no terms proportional to [itex](\partial_{x}\psi)^{2}[/itex] or something like that.

    In the case of time-dependent SE for one 1D particle, we have [itex]d(x,t)=i\hbar[/itex], [itex]b(x,t)=\frac{\hbar^{2}}{2m}[/itex], [itex]f(x,t)=-V(x,t)[/itex] and other coefficients are zero.
     
    Last edited: Jul 21, 2013
  4. Jul 21, 2013 #3

    jtbell

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    Assume that ##\psi_1## and ##\psi_2## each satisfy the SE. Substitute the linear combination ##a\psi_1 + b\psi_2## (where a and b are constants) into the SE and show that it is also a solution.
     
  5. Jul 22, 2013 #4

    tom.stoer

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    The time-dep. SE can be written as

    ##(i\partial_t - H)\psi = 0##

    So if you have two solutions i=1,2 then any linear combination is a solution, too


    ##(i\partial_t - H)\psi_i = 0 \;\to\; (i\partial_t - H)(a_1\psi_1 + a_2\psi_2) = 0##
     
  6. Jul 22, 2013 #5

    bhobba

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    Actually you have got the cart before the horse. We know the superposition principle is valid because the states form a Hilbert space. That time evolution is linear follows from Wigners theorem and the invariance of the Born Rule:
    http://en.wikipedia.org/wiki/Wigner's_theorem.

    Like so much in modern physics its deep reason lies in symmetry which is itself something very deep and interesting. See the following I read ages ago at the library and have been meaning to get a copy myself:
    https://www.amazon.com/Symmetries-R...=sr_1_1?s=books&ie=UTF8&qid=1374543605&sr=1-1

    You will find a proper axiomatic development of QM and see what is basic and what isn't in Ballentine's - QM - A Modern Development. There is really only two axioms - the first is about the eigenvalues of operators on a Hilbert Space (that's the reason the superposition principle holds - its a basic property of Hilbert Spaces - but the exact reasoning is interesting and slightly subtle - get the book to see what I mean - it's got to do with so called pure and mixed states), the second is the Born Rule. And from Gleason's Theorem the second follows from the first with a very reasonable assumption so one could argue it really involves just one axiom - strictly speaking it doesn't - you need the second axiom - but the assumption (non contextuality) is just so reasonable at the mathematical level in light of the first.

    Thanks
    Bill
     
    Last edited by a moderator: May 6, 2017
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