Quick question on Probability Current

[Seydlitz]

Here's the probability current equation as seen in Griffith's book.

$$j = \frac{\hbar}{2mi}\left(\Psi^* \frac{\partial \Psi }{\partial x}- (\frac{\partial \Psi^* }{\partial x})\Psi \right)$$

Does the second right term instruct us to take the derivative of the wave function complex conjugate? Or for that matter $$\Psi^*$$ this refers to the complex conjugate right?

Secondly, what math topics are included in Schrödinger's equation? Is it only differential calculus or does it also include complex variable calculus?

Thank You

 

[vanhees71]

Since $t \in \mathbb{R}$, of course
$$\frac{\partial \psi^*}{\partial t}=\left (\frac{\partial \psi}{\partial t} \right )^*.$$
In principle the Schrödinger equation is a partial differential equation of real arguments for a complex-valued function.

Of course, as always, the use of complex analysis is of great advantage to find solutions. The masters of this approach were Sommerfeld and Pauli. You can read about this in Pauli's marvelous lectures on the subject.

 

[Seydlitz]

Since $t \in \mathbb{R}$, of course
$$\frac{\partial \psi^*}{\partial t}=\left (\frac{\partial \psi}{\partial t} \right )^*.$$
In principle the Schrödinger equation is a partial differential equation of real arguments for a complex-valued function.

Of course, as always, the use of complex analysis is of great advantage to find solutions. The masters of this approach were Sommerfeld and Pauli. You can read about this in Pauli's marvelous lectures on the subject.

Ok thank you for the information. Should I read some math books about complex analysis or does the basic technique usually discussed in normal calculus books? Where can you watch Pauli's lecture on this?