Quick question on Probability Current

In summary, the probability current equation, as described in Griffith's book, involves taking the derivative of the wave function complex conjugate. The Schrödinger equation includes topics from differential and complex variable calculus, and the use of complex analysis is often advantageous in finding solutions. Masters of this approach include Sommerfeld and Pauli, whose lectures on the subject can be found in Pauli's writings.
  • #1
Seydlitz
263
4
Here's the probability current equation as seen in Griffith's book.

[tex]j = \frac{\hbar}{2mi}\left(\Psi^* \frac{\partial \Psi }{\partial x}- (\frac{\partial \Psi^* }{\partial x})\Psi \right)[/tex]

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 Schrodinger's equation? Is it only differential calculus or does it also include complex variable calculus?

Thank You
 
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  • #2
Since [itex]t \in \mathbb{R}[/itex], of course
[tex]\frac{\partial \psi^*}{\partial t}=\left (\frac{\partial \psi}{\partial t} \right )^*.[/tex]
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.
 
  • #3
vanhees71 said:
Since [itex]t \in \mathbb{R}[/itex], of course
[tex]\frac{\partial \psi^*}{\partial t}=\left (\frac{\partial \psi}{\partial t} \right )^*.[/tex]
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?
 

What is probability current?

Probability current is a concept in quantum mechanics that describes the flow of probability density through a particular region of space. It is a measure of the rate at which a particle is likely to be found in a given area over time.

How is probability current calculated?

The probability current is calculated by taking the product of the probability density and the velocity of the particle. This can be written mathematically as J = ρv, where J is the probability current, ρ is the probability density, and v is the velocity of the particle.

What is the significance of probability current?

Probability current is important in quantum mechanics because it helps us understand the behavior of particles at the quantum level. It allows us to predict the movement and behavior of particles, and is a key concept in many quantum mechanical equations and theories.

Can probability current be negative?

Yes, probability current can be negative. This indicates that there is a flow of probability density in the opposite direction of the particle's velocity. This can happen in situations where particles are moving in opposite directions or when there is interference between particles.

How is probability current related to current in classical physics?

Probability current is similar to current in classical physics in that it represents the flow of a quantity over time. However, in classical physics, current refers to the flow of charge, while in quantum mechanics, probability current refers to the flow of probability density. Additionally, probability current can have negative values, whereas current in classical physics is always positive.

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