I am studying the property of Schrödinger equation that it automatically preserves the normalization of the wave function. We know:

$$\int_{-\infty}^\infty \frac{\partial}{\partial t}|\Psi(x,t)|^2dx$$

By product rule:

$$\frac{\partial}{\partial t}|\Psi|^2 = \Psi^*\frac{\partial\Psi}{\partial t} + \Psi\frac{\partial\Psi^*}{\partial t}$$

Now I want to determine the value of $\frac{\partial\Psi}{\partial t}$ and $\frac{\partial\Psi^*}{\partial t}$ and to do so we just have to use the Schrödinger equation:

$$i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi$$

My book (Introduction to QM by David J. Griffiths) states:

$$\frac{\partial\Psi}{\partial t} = \frac{i\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2} - \frac{i}{\hbar}V\Psi$$

But I got:

$$\frac{\partial\Psi}{\partial t} = -\frac{i\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2} + \frac{i}{\hbar}V\Psi$$

The same happened to me with the complex conjugate. Is this a mistake from the book or mine?


I'm afraid it's your mistake! $i^2=-1$ so dividing by $i$ is the same as multiplying by $-i$.


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