Derivation of "continuity equation"

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SUMMARY

The continuity equation in Quantum Mechanics can be derived using the Schrödinger equation. Begin by multiplying the Schrödinger equation for the wave function \(\Psi(\vec{x},t)\) by its complex conjugate \(\Psi^*(\vec{x},t)\), and then do the reverse for \(\Psi^*(\vec{x},t)\). Subtracting these two results allows for the definition of probability density and probability current density, leading to the final form of the continuity equation. This method is effective and has been confirmed by users in the discussion.

PREREQUISITES
  • Understanding of the Schrödinger equation in Quantum Mechanics
  • Familiarity with complex conjugates in wave functions
  • Knowledge of probability density and current density concepts
  • Basic mathematical manipulation skills
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  • Study the derivation of the Schrödinger equation in detail
  • Learn about probability density and current density in Quantum Mechanics
  • Explore applications of the continuity equation in physical systems
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Students and researchers in Quantum Mechanics, physicists focusing on wave functions, and anyone interested in the mathematical foundations of quantum theories.

Ami
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Hello,

I need the derivation of "continuity equation" by the current density equation,in Quantum Mechanics.

I really need this derivation quickly,please

Thanks
 
Last edited:
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You can do it yourself.

1. Take the Schrödinger equation for \Psi(\vec{x},t) and multiply it by \Psi^*(\vec{x},t).
2. Take the Schrödinger equation for \Psi^*(\vec{x},t) and multiply it by \Psi(\vec{x},t).
3. Subtract the two equations.

From there you have to appropriately define the probability density and the probability current density, then finish off the manipulations to get the continuity equation. Try that, and if you get stuck post what you have done and we will get you un-stuck.
 
Thank you very much for your reply

It has been very helpful to me.
And I've finished deriving the equation.

Thanks a lot
 
Last edited:

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