derive the equation for wave function of the wave.
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Answer:
One may derive the Schrödinger equation starting from the Dirac-von Neumann axioms. Suppose the wave function {\displaystyle \psi (t_{0})}{\displaystyle \psi (t_{0})} represents a unit vector defined on a complex Hilbert space at some initial time {\displaystyle t_{0}}t_{0}. The unitarity principle requires that there must exist a linear operator, {\displaystyle {\hat {U}}(t)}{\displaystyle {\hat {U}}(t)}, such that for any time {\displaystyle t>t_{0}}{\displaystyle t>t_{0}},
{\displaystyle \psi (t)={\hat {U}}(t)\psi (t_{0}).}{\displaystyle \psi (t)={\hat {U}}(t)\psi (t_{0}).}
(1)
Given that {\displaystyle \psi (t)}\psi(t) must remain a unit vector, the operator {\displaystyle {\hat {U}}(t)}{\displaystyle {\hat {U}}(t)} must therefore be a unitary transformation. As such, there exists an exponential map such that {\displaystyle {\hat {U}}(t)=e^{-i{\hat {\mathcal {H}}}t}}{\displaystyle {\hat {U}}(t)=e^{-i{\hat {\mathcal {H}}}t}} where {\displaystyle {\hat {\mathcal {H}}}}{\displaystyle {\hat {\mathcal {H}}}} is a Hermitian operator (given by the fact that the Lie algebra of the unitary group is generated by skew-Hermitian operators, and if {\displaystyle {\hat {\mathcal {H}}}}{\displaystyle {\hat {\mathcal {H}}}} is Hermitian, then {\displaystyle -i{\hat {\mathcal {H}}}}{\displaystyle -i{\hat {\mathcal {H}}}} is skew-Hermitian). Therefore, the first-order Taylor expansion of {\displaystyle {\hat {U}}(t)}{\displaystyle {\hat {U}}(t)} centered at {\displaystyle t_{0}}t_{0} takes the form
{\displaystyle {\hat {U}}(t)\approx 1-i(t-t_{0}){\hat {\mathcal {H}}}.}{\displaystyle {\hat {U}}(t)\approx 1-i(t-t_{0}){\hat {\mathcal {H}}}.}
And so, substituting the above expansion into (1) thus yields
{\displaystyle \psi (t)=\psi (t_{0})-i(t-t_{0}){\hat {\mathcal {H}}}\psi (t_{0}),}{\displaystyle \psi (t)=\psi (t_{0})-i(t-t_{0}){\hat {\mathcal {H}}}\psi (t_{0}),}
which, when rearranged and taken in the limit {\displaystyle t\rightarrow t_{0}}{\displaystyle t\rightarrow t_{0}}, provides an equation of the same form as the Schrödinger equation
{\displaystyle i{\frac {d\psi }{dt}}={\hat {\mathcal {H}}}\psi ,}{\displaystyle i{\frac {d\psi }{dt}}={\hat {\mathcal {H}}}\psi ,}
where the ordinary definition for the derivative was used. However, the operator {\displaystyle {\hat {\mathcal {H}}}}{\displaystyle {\hat {\mathcal {H}}}} used here denotes an arbitrary Hermitian operator. Nonetheless, by using the correspondence principle it is possible to show that in the classical limit, using appropriate units, the expectation value of {\displaystyle {\hat {\mathcal {H}}}}{\displaystyle {\hat {\mathcal {H}}}} indeed corresponds to the Hamiltonian of the system.[7]