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Discussion of the Particle in a Box Problem

If we were looking at a comparable Classical Mechanics problem we could drop a marble into tall box just wide enough in one direction to accomodate the marble but of width 'L' in the other horizontal direction.

How would we expect the marble to behave?

We would expect the marble to drop to the bottom of the box with a thunk, make a few small bounces, perhaps roll along the bottom briefly, and then come to rest.

  • The gravitational potential energy which the marble intially possessed would be converted into kinetic energy as it picked up speed falling downward.
  • Some kinetic energy would be converted into heat as the marble made a semi-inelastic collision with the bottom of the box.
  • The remainder of the energy would be converted back into both gravitational potential and kinetic energy as it bounced upwards.
  • This process would repeat itself until at last all of the potential and kinetic energy becomes dissipated as heat, and as the marble assumes a zero energy state sitting still at the bottom of the box.

How would we expect a quantum particle to behave?

We know from the initial conditions that the quantum partical's potential energy was zero. So there could no unbalanced gravitational pull on our particle, but it could have variable amounts of kinetic energy as it bounces back and forth between the sides of the square well.

However there will be some odd behaviours in the particle.
  • The speed of the particle cannot decline gradually the way the marble did, it would need to decline in steps which would correspond to the specific allowable energy levels determined by the valid solutions we found to Schrodinger Equation. Therefore the energy transfer process to its environment could not be a continuous process but rather it would be lumpy, or bumpy, as you prefer.
  • The marble would change from one velocity to another suddenly and the energy transfer to the environment would be similarly sudden, perhaps even explosive in nature.
  • Oddly, even though the speed of the particle must be kept constant, our chances of finding the particle at specific locations along the bottom of the well is lumpy. The slower the particle moves (e.g the less energetic the particle is), the lumpier the postion distribution looks.
  • Finally whereas the marble came to rest, our quantum particle will never completely come to rest since, as we will learn later the zero energy state is not an allowed state.

So now we see why this is called quantum mechanics. The energy shifts in the quantum system can only take place in discrete steps or quanta.

The tranfer of energy from the system to the environment takes place as discrete events and the energy transfered can only be of certain predefined amounts characterized by the energy difference between allowed energy states of the particle.

It is instructive to note that the energy quantization requirement is imposed and characterized by the action of physically constraining the particle to a region 0 to L. If there had been no physical constraints put on the location of the particle there would have been no need to enforce the boundary conditions which resulted in the quantized energy levels. If the size of the well were different, so to would be the allowed energy levels,

Now you are beginning to have an inkling of what it would be like to live life as a quantum particle and how the rules of the quantum world differ from those of our everyday classical mechanical world.

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