Imagine a particle of mass m, constrained to move along the x-axis, subject to some specified force F(x, t). The program of classical mechanics is to deter- mine the position of the particle at any given time: x(t). Once we know that, we can figure out the velocity (\( v=\frac{dx}{dt}\) ), the momentum (p = mv), the kinetic energy ( \( T=\frac{1}{2}mv^2 \) ), or any other dynamical variable of interest. And how do we go about determining x(t)? We apply Newton's second law: F = ma. (For conservative systems the only kind we shall consider, and, fortunately, the only kind that occur at the microscopic level---the force can be expressed as the derivative of a potential energy function, \( F=-\frac{\partial V}{\partial x} \) , and Newton's law reads \( m\frac{d^2x}{dt^2}=-\frac{\partial V}{\partial x} \) .) This, together with appropriate initial conditions (typically the position and velocity at t 0), determines x(t). Quantum mechanics approaches this same problem quite differentl...
Imagine a particle of mass m, constrained to move along the x-axis, subject to some specified force F(x, t). The program of classical mechanics is to deter- mine the position of the particle at any given time: x(t). Once we know that, we can figure out the velocity (\( v=\frac{dx}{dt}\) ), the momentum (p = mv), the kinetic energy ( \( T=\frac{1}{2}mv^2 \) ), or any other dynamical variable of interest. And how do we go about determining x(t)? We apply Newton's second law: F = ma. (For conservative systems the only kind we shall consider, and, fortunately, the only kind that occur at the microscopic level---the force can be expressed as the derivative of a potential energy function, \( F=-\frac{\partial V}{\partial x} \) , and Newton's law reads \( m\frac{d^2x}{dt^2}=-\frac{\partial V}{\partial x} \) .) This, together with appropriate initial conditions (typically the position and velocity at t 0), determines x(t).
Quantum mechanics approaches this same problem quite differently. In this case what we're looking for is the wave function, ψ(x, t), of the particle, and we get it by solving the Schrödinger equation:
\( i\hbar\frac{\partial \Psi}{\partial t} = -\frac{\hbar ^2}{2m} \frac{\partial ^2 \Psi}{\partial x^2} +V\Psi \)
Here i is the square root of -1, and ħ is Planck's constant or rather, his original constant (h) divided by 2π :
\( \hbar = \frac{h}{2\pi} = 1,054573 \times 10^{-34} \) Js
The Schrödinger equation plays a role logically analogous to Newton's second law: Given suitable initial conditions (typically, ψ(x, 0)), the Schrödinger equation determines (x, t) for all future time, just as, in classical mechanics, Newton's law determines x(t) for all future time.
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