Q:

How can one derive Schrödinger equation?

The Schrödinger equation is the basis to understanding quantum mechanics, but how can one derive it? I asked my instructor but he told me that it came from the experience of Schrödinger and his experiments. My question is, can one derive the Schrödinger equation mathematically?

A:

Be aware that a "mathematical derivation" of a physical principle is, in general, not possible. Mathematics does not concern the real world, we always need empirical input to decide which mathematical frameworks correspond to the real world.
However, the Schrödinger equation can be seen arising naturally from classical mechanics through the process of quantization. More precisely, we can motivate quantum mechanics from classical mechanics purely through Lie theory, as is discussed here, yielding the quantization prescription
$$ \{\dot{},\dot{}\} \mapsto \frac{1}{\mathrm{i}\hbar}[\dot{},\dot{}]$$
for the classical Poisson bracket. Now, the classical evolution of observables on the phase space is 
$$ \frac{\mathrm{d}}{\mathrm{d}t} f = \{f,H\} + \partial_t f$$
and so its quantization is the operator equation
$$ \frac{\mathrm{d}}{\mathrm{d}t} f = \frac{\mathrm{i}}{\hbar}[H,f] + \partial_t f$$
which is the equation of motion in the Heisenberg picture. Since the Heisenberg and Schrödinger picture are unitarily equivalent, this is a "derivation" of the Schrödinger equation from classical phase space mechanics.

A:

Small addition to ACuriousMind's great answer, in reply to some of the comments asking for a derivation of Schrödinger wave equation, using the results of Feynman's path integral formalism:
(Note: not all steps can be included here, it would be too long to remain in the context of a forum-discussion-answer.)
In the path integral formalism, each path is attributed a wavefunction $\Phi[x(t)]$, that contributes to the total amplitude, of let's say, to go from $a$ to $b.$ The $\Phi$'s have the same magnitude but have differing phases, which is just given by the classical action $S$ as was defined in the Lagrangian formalism of classical mechanics.  So far we have: $$ S[x(t)]= \int_{t_a}^{t_b} L(\dot{x},x,t) dt $$
and $$\Phi[x(t)]=e^{(i/\hbar) S[x(t)]}$$
Denoting the total amplitude $K(a,b)$, given by: $$K(a,b) = \sum_{paths-a-to-b}\Phi[x(t)]$$
The idea to approach the wave equation, describing the wavefunctions as a function of time, we should start by dividing the time interval between $a$-$b$ into $N$ small intervals of length $\epsilon$, and for a better notation, let's use $x_k$ for a given path between $a$-$b$, and denote the full amplitude, including its time dependance as $\psi(x_k,t)$ ($x_k$ taken over a region $R$): 
$$\psi(x_k,t)=\lim_{\epsilon \to 0} \int_{R} \exp\left[\frac{i}{\hbar}\sum_{i=-\infty}^{+\infty}S(x_{i+1},x_i)\right]\frac{dx_{k-1}}{A} \frac{dx_{k-2}}{A}... \frac{dx_{k+1}}{A} \frac{dx_{k+2}}{A}... $$
Now consider the above equation if we want to know the amplitude at the next instant in time $t+\epsilon$: 
$$\psi(x_{k+1},t+\epsilon)=\int_{R} \exp\left[\frac{i}{\hbar}\sum_{i=-\infty}^{k}S(x_{i+1},x_i)\right]\frac{dx_{k}}{A} \frac{dx_{k-1}}{A}... $$
The above is similar to the equation preceding it, the difference relying on the hint that, the added factor with $\exp(i/\hbar)S(x_{k+1},x_k)$ does not involve any of the terms $x_i$ before $i<k$, so the integration can be preformed with all such terms factored out. All this reduces the last equation to:
$$\psi(x_{k+1},t+\epsilon)=\int_{R} \exp\left[\frac{i}{\hbar}\sum_{i=-\infty}^{k}S(x_{i+1},x_i)\right]\psi(x_k,t)\frac{dx_{k}}{A}$$
Now a quote from Feynman's original paper, regarding the above result: 

This relation giving the development of $\psi$ with time will be
  shown, for simple examples, with suitable choice of $A$, to be
  equivalent to Schroedinger's equation. Actually, the above equation is
  not exact, but is only true in the limit $\epsilon \to 0$ and we shall
  derive the Schroedinger equation by assuming this equation is valid
  to first order in $\epsilon$. The above need only be true for small
  $\epsilon$ to the first order in $\epsilon.$

In his original paper, following up the calculations for 2 more pages, from where we left things, he then shows that: 

Canceling $\psi(x,t)$ from both sides, and comparing terms to first
  order in $\epsilon$ and multiplying by $-\hbar/i$ one obtains

$$-\frac{\hbar}{i}\frac{\partial \psi}{\partial t}=\frac{1}{2m}\left(\frac{\hbar}{i}\frac{\partial}{\partial x}\right)^2 \psi + V(x) \psi$$ which is Schroedinger's equation.
I would strongly encourage you to read his original paper, don't worry it is really well written and readable.

References: 
Space-Time Approach to Non-Relativistic Quantum Mechanics by R. P. Feynman, April 1948.
Feynman Path Integrals in Quantum Mechanics, by Christian Egli

A:

According to Richard Feynman in his lectures on Physics, volume 3, and paraphrased "The Schrodinger Equation Cannot be Derived". According to Feynman it was imagined by Schrodinger, and it just happens to provide the predictions of quantum behavior. 

