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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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  • OP's proposal (v5) seems to be just notation with no changes to physics per se, and hence mainly opinion-based. – Qmechanic Feb 23 '23 at 09:36

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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    Thanks, it appears to me now except Schrödinger equation all other physical quantities retain their conventional meaning. – Narasimham Feb 22 '23 at 23:10
  • What do you mean ? All physical constants, including $\hbar$ are real, there is nothing unconventional here. What may appear new to you in Schroedinger's equation is the appearance of complex numbers, but this is one of the requirements of the description of quantum mechanics. – Adrien Martina Feb 22 '23 at 23:15
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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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