Deducshre

Deducshre is deduction of the Schroedinger equation from the basic principles of Quantum mechanics. The detailed deduction, with all the intermediate formulas, is suggested below.

Consider the Schroedinger equation in the following form:

\((1) ~ ~ ~ \displaystyle \hbar \mathrm i \frac{\mathrm d \psi}{\mathrm d t}= H \psi \)

The goal is, by the set of transforms, to convert this equation to identity. Then repetition of the inverse transforms in the inverse order provides the proof of equation (1).

First of all, cancel \(\mathrm d\) in the numerator and the denominator of the fraction in left hand side of (1). This gives

\((2) ~ ~ ~ \displaystyle \hbar \mathrm i \frac{\psi}{t}= H \psi \)

Cancel \(\psi\) in the left hand side and in the right hand side of equation (2), this gives

\((3) ~ ~ ~ \displaystyle \hbar \mathrm i \frac{1}{t}= H \)

Multiply both sides of equaiton (3) by \(t\); this gives

\((4) ~ ~ ~ \displaystyle \hbar \mathrm i = H t \)

Nothe that \(\hbar\) is just "h bar", and \(H\) is just \(h\) capital.

Due to the associativity, that is fundamental principle of Quantum Mechanics, in the left hand side of (4), operation "bar" can be applied to \(\mathrm i\) instead of \(h\), converting it to \(T\). In this way, we get \(hT\) in the left hand side of (4).

In the similar way, in the right hand side of (4), instead of to apply operaiton "Capital" to \(h\), it can be applied to \(t\) in the following way:

\((5) ~ ~ ~ \displaystyle h T= h \mathrm{Capital} t \)

Obviously, \(~\mathrm{Capital} t=T~ ~\) . So, equation (5) becomes identity.

Unitary evolution is also principal postulate of Quantum mechanics. Any unitary evolution allows inversion. So, the evolution (1)–(5) can be inverted; that leads from identity (5) to the Shcroedinger equation (1).

(End of proof)

Hisory

The elegant proof above should be attributed to Roger Hermann, who had suggested it in centiry 20 – the same century, when famous Erwin Scroedinger had wrote his famous Schroedinger equation (1).

Keywords

Schroedinger equation, Roger Herrmann, Unitary evolution, Associativity, Quantum mechanics