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Physics, Astronomy & Cosmology / Re: Complex or real wave function?
« on: 15/11/2022 12:18:14 »
Hi.
Also, it's a probability density rather than a probability on it's own. Given a small element of space δx and of time δt then you do need to multiply by the space-time volume element δt δx, so that |ψ(x,t)|2 . δx . δt = probability of finding the particle in the element of space between x and x + δx and at a time between t and t + δt.
Similar idea if it's 3-dimensional space instead of just one dimensional along the x-axis. Then we use the space element δ3x = δx δy δz.
Anyway, the first part of what CompuAI had written looks OK. You CAN consider the Schrodinger equation as a some differential equations involving only a real valued function(s) if you want to. In that case, you have a PAIR of simulatenous differential equations linking time derivatives of Ψ1 (x,t) to space derivatives of Ψ2 (x,t) (and vice versa.... time derivatives of Ψ2 are linked to space derivatives of Ψ1 ) but both Ψ1 and Ψ2 are real-valued functions.
In general, it's NOT easier that way. In solving that pair of simultaneous differential equations you probably would start by just writing it as one combined differential equation with a complex valued function. However, it's of some relevance because there is always the discussion point about whether Complex numbers are in some sense "real", i.e. that they have some deep real world meaning because they are essential in Quantum Mechanics. The sort of thing where you replace the Schrodinger equation with a pair of real valued differential equations suggests that, no you don't need Complex numbers, you can keep all wave functions real valued BUT it's harder that way.
Best Wishes.
As I understand it, the magnitude of a wave function indicates the probability of detecting the particle in a particular place.Yes. At a particular place and also a particular time if ψ = ψ(x,t) = a function of space AND time, which is how CompuAI seemed to be representing the wave function.
Also, it's a probability density rather than a probability on it's own. Given a small element of space δx and of time δt then you do need to multiply by the space-time volume element δt δx, so that |ψ(x,t)|2 . δx . δt = probability of finding the particle in the element of space between x and x + δx and at a time between t and t + δt.
Similar idea if it's 3-dimensional space instead of just one dimensional along the x-axis. Then we use the space element δ3x = δx δy δz.
Anyway, the first part of what CompuAI had written looks OK. You CAN consider the Schrodinger equation as a some differential equations involving only a real valued function(s) if you want to. In that case, you have a PAIR of simulatenous differential equations linking time derivatives of Ψ1 (x,t) to space derivatives of Ψ2 (x,t) (and vice versa.... time derivatives of Ψ2 are linked to space derivatives of Ψ1 ) but both Ψ1 and Ψ2 are real-valued functions.
In general, it's NOT easier that way. In solving that pair of simultaneous differential equations you probably would start by just writing it as one combined differential equation with a complex valued function. However, it's of some relevance because there is always the discussion point about whether Complex numbers are in some sense "real", i.e. that they have some deep real world meaning because they are essential in Quantum Mechanics. The sort of thing where you replace the Schrodinger equation with a pair of real valued differential equations suggests that, no you don't need Complex numbers, you can keep all wave functions real valued BUT it's harder that way.
Best Wishes.
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