Quantum Wave in a Box на пк

1. - Watch both solution ψ(x,t) and free wave-packet curves evolve together in time and separate when entering non-zero potential energy region.

2. The solution ψ(x,t) of the time-dependent Schrödinger equation is then computed as ψ(x,t) = exp(-iHt) ψ₀(x) where ψ₀(x) is a gaussian wave-packet at initial time t = 0.

3. - Change initial gaussian parameters of the wave-packet (position, group velocity, standard deviation), enter any time value, then tap refresh button to observe changes in curves without new diagonalization.

4. - Spatially continuous problem discretized over [a, b] and time-independent Schrödinger equation represented by a system of N+1 linear equations using a 3, 5 or 7 point stencil; N being the number of x-steps.

5. - Animation shows gaussian wave-packet ψ(x,t) evolving with real-time evaluation of average velocity, kinetic energy and total energy.

6. A solution of this differential equation represents the motion of a non-relativistic particle in a potential energy field V(x).

7. The time-independent Schrödinger equation H ψ(x) = E ψ(x), represented by a set of linear equations, is solved by using quick diagonalization routines.

8. - Quantum system defined by mass, interval [a, b] representing the Box and (real) potential energy V(x).

9. In Quantum Mechanics the one-dimensional Schrödinger equation is a fundamental academic though exciting subject of study for both students and teachers of Physics.

10. This is particularly useful to get a (usually more precise) solution for any time value t when animation is slower in cases of N being large.

11. Actually the originally continuous x-spatial differential problem is discretized over a finite interval (the Box) while time remains a continuous variable.

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