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Trying to get a first understanding of QM. The Schrödinger equation in standard form for $\Psi$

$$ i \hbar\frac{\partial }{\partial t} \Psi(x,t) =\left[-\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial t^2} +V(x,t)\right]$$

Can we look at it this way, since we can have both signs for $ i= \pm \sqrt{-1} $ and agree to accommodate/use Planck's constant also as an imaginary constant $i \hbar\to \hbar $ can the following Schrödinger equation form still interpret or represent negative potential energy $V$ (unconventionally) for same wave function $\Psi?$

$$ \hbar \frac{\partial }{\partial t} \Psi(x,t) =\left[\frac{\hbar ^2}{2m}\frac{\partial^2 }{\partial t^2}+V(x,t)\right].$$

An advantage could be that an imaginary quotient need explicitly occur in the PDE. I am not sure of the constants making sense.

Qmechanic
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Narasimham
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3 Answers3

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Redefining Planck's constant $\hbar \to i\hbar$ wouldn't be restricted to Schrödinger's equation, because Schrödinger's equation is by far not the only equation containing $\hbar$. There are many other equations currently containing $\hbar$ but no $i$, for example: $$\begin{align} &\text{Energy: } &E&=\hbar\omega \\ &\text{Momentum: } &\vec{p}&=\hbar\vec{k} \\ &\text{Spin: } &\vec{S}&=\frac 12\hbar\vec{\sigma} \\ &... \end{align}$$

In other words: Replacing $\hbar \to i\hbar$ would create a whole mess by inserting an $i$ into the equations above where we currently don't have it.

Thomas Fritsch
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We prefer real numbers to imaginary numbers for our physical constants. Therefore if $\hbar$ were to be purely imaginary, we would rather call it $i\hbar$ to make it real.

So I assume your question is rather, "Why do we need complex coefficients in the Schroedinger equation ?". The answer to this question is that if we removed the $i$, the total probability of the particle being at any position $x$ would not stay being 1, but would change over time.

I redirect you to this post : What do imaginary numbers practically represent in the Schrödinger equation?

Adrien Martina
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The change $\hbar \rightarrow ih \;\;$is equivalent to $\;\;i(i\hbar)\frac{\partial \psi}{\partial it}=H\psi$, time becomes imaginary and the equation becomes the heat equation.

.... the Schrödinger equation of quantum mechanics can be regarded as a heat equation in imaginary time.

The Tiler
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