We know that superposition principle is valid in Quantum Mechanics

[dorazyl]

because Schrödinger equation is a linear differential equation. How do we show that Schrödinger equation is a linear differential equation?

 

[hilbert2]

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

$F(\partial_{tt}\psi,\partial_{xx}\psi,\partial_{xt}\psi,\partial_{t} \psi,\partial_{x}\psi,\psi)=g(x,t)$

If the equation is linear, then $F$ is linear with respect to all its arguments, i.e.

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

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

In the case of time-dependent SE for one 1D particle, we have $d(x,t)=i\hbar$, $b(x,t)=\frac{\hbar^{2}}{2m}$, $f(x,t)=-V(x,t)$ and other coefficients are zero.

 

[jtbell]

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.

 

[tom.stoer]

How do we show that Schrödinger equation is a linear differential equation?

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


$(i\partial_t - H)\psi_i = 0 \;\to\; (i\partial_t - H)(a_1\psi_1 + a_2\psi_2) = 0$

 

[bhobba]

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/dp/0918024161/?tag=pfamazon01-20

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