Undergrad Probability current density

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Calculating the probability density |ψ(x)|² is sufficient to determine if a quantum particle crosses a barrier. However, probability current density is essential for understanding the conservation of probability, particularly in scenarios involving boundaries. While it is often overlooked, current density provides insight into the flow of probability, which is crucial for a complete analysis. The discussion emphasizes that while probability density can indicate presence, current density is necessary for a deeper understanding of quantum behavior. Thus, both concepts serve distinct but important roles in quantum mechanics.
Abdul Quader
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In order to check if a quantum particle crosses a barrier or not, isn't calculating ##|\psi(x)|^2## enough in that particular region ? Why do we need to calculate probability current density for that matter?
 
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It is not needed for this, and it is usually never computed. It is needed in the proof that probability is conserved unless it flows into or out of a boundary.
 
Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?

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