Calculating Probability Density in Quantum Mechanics

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In quantum mechanics, the probability density is calculated using the absolute square of the wave function, ψ, represented as (mod[ψ]^2)*dx*dy*dz, which reflects the likelihood of finding a particle in a specific space at a given instant. The suggestion to include a time differential element, dt, in the calculation is considered unnecessary since probability density pertains to spatial distribution at a fixed moment. The integral of the probability density over a spatial interval yields the probability of locating the particle within that space at that instant. Clarifications emphasize that ψ itself represents the probability density rather than being a variable used to compute it. The discussion highlights a misunderstanding of the fundamental concepts in quantum mechanics regarding the nature of probability density.
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In quantum mechanics, given ψ, we calculate the probability density in a given space by (mod[ψ]^2)*dx*dy*dz (as given in any standard textbook). But my suggetion is that it should be (mod[ψ]^2)*dx*dy*dz*dt where dt is the differential time element. Its seems to me to be more fundamental. Then why don't we do that?
Thanks
Curious undergrad!
 
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The probability density is at a given instant, not per unit time.
 
Also we do NOT use \Psi to calculate the probability density. \Psi is the probability density. The integral, over a space interval, gives the probability that the object is in that space at the given instant.
 
@HallsofIvy I think you mean the absolute square of the wave function is the probability density.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

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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