How Newton's law of motion can be derived from Schrodinger's equations?
How velocity and acceleration expectation values can be calculated from Schrodinger's equation, like position expectation value is position times probability density amplitude squared over whole space?
Read in Wikipedia the theorems of Ehrenfest, and you will see the answer.
Dear Sumit Bhowmick ,
If you mean by "reduce" that:
Is there in quantum mechanics a law similar to " Newton's law of motion"?
So the answer is as Sofia D. Wechsler answered you.
but if you mean:
Can we deduce the classical mechanics from quantum mechanics?
As Bohr said that quantum mechanics does not produce classical mechanics in a similar way as classical mechanics arises as an approximation of special relativity at velocities very slow than light speed.
He argued that classical mechanics exists independently of quantum mechanics and cannot be derived from it.
Max Jammer has said: ”quantum mechanics and classical dynamics are built on fundamentally different foundations”!
Only the correspondence principle states that the behavior of systems described by quantum mechanics reproduces in a statistical way the classical mechanics in the limit of large quantum numbers
With best regards.
Newton's equation describes a physical system and can predict how this changes with a set of initial conditions, we can solve for the force, mass and acceleration given enough information.
The Schrodinger equation also describes a physical system where we solve for the azimuth, polar and radial points of a spherical coordinate system (r, θ, ϕ). The difficulty mostly is in the conversion from three-dimensional space (x, y, z) to spherical coordinates within the second-order partial derivatives. The equation then needs to be separated into two or three parts requiring several algebraic methods. The inputs also include the quantum numbers n, l and m.
When dividing the equation into two parts of the radial component (r) and the angular component (θ, ϕ). For the radial solution we have two parts of the expression: the exponential term that signifies attraction to the nucleus and the polynomial term signifying electron shell harmonics. The polynomial term is solved with the Laguerre polynomial and then scaling factors are used to solve the radial wave function.
The angular component (θ, ϕ) is solved again with scaling factors, normalization and then the Legendre polynomial. The arrays of the radial solution and the angular solution are then multiplied for solving the wave function.
The three solution equations of the spherical coordinate system are then applied to the spherical coordinate space. These equations and quantum conditions describe the orbital shape of the system. For instance the most simple system is that of the hydrogen atom and therefore its orbital shape of the electron, knowing it has one electron and one proton.
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