The following are the subtle differences that separate Quantum Mechanics from Classical Mechanics.

1. Transmission: Particles are reflected that classically should not.

2. Tunneling: Particles pass through potential barriers.

3. Discrete Energy Values: Energy states exist at discrete levels as opposed to continuous energy values. An example is the harmonic oscillator, which has energy levels at hw/2, 3hw/2, 5hw/2....

4. E¹0: The lowest energy value is not zero. Again, the harmonic oscillator serves as an example. E₀= hw/2.

5. Scatter Out of a Finite Well: Particles can scatter out of a potential well that classical particles would normally remain in.

6. The Uncertainty Principle: You cannot measure x and p with arbitrary precision at the same time. You can only know one at a time.

sx sp=h/2

or in a more formal language

[x,p]=ih ¹ 0

This happens because the QM momentum operator P=-ih(¶/¶x).

7. Spin: Classical particles have a form of angular momentum analogous to classical spin. Quantum spin, like other Quantum values, exists in discrete values. In attempt to explore the z component of an electron, Stern and Gerlach, passed silver atoms through a magnetic field. The stream was separated into two distinct groupings which can be regarded as spin up and spin down. A classical particle would have a continuous spectrum. The most important spin value is spin ½ which is found in electrons, protons, and neutrons.

8. Identical Particles: Classically two particles are distinguishable and will exist in separate states. Particle r₁ would occupy ya and particle r₂ would occupy yb, which would be written as y(r₁, r₂)= y_a(r₁) y_b(r₂). But in a Quantum system, one particle would occupy y_a and the other y_b but we cannot say which one is in which. We would write this as y(r₁, r₂)= y_a(r₁) y_b(r₂) ± y_a(r₂) y_b(r₁). The ± refers to bosons, which have integer value spins and are therefore added, and fermions which have half spin and are subtracted. A direct result of this flag is the Pauli Exclusion principle that states that two fermions cannot exist in the same state.

9. Exchange Force: The exchange force is not a force in the literal sense. It is the result of the expected locations of fermions and bosons. In the classical world two particles would have a distance between them of d. Yet in the quantum world, fermions are repelled apart by the exchange force to a distance greater than d. Bosons seem attracted to each other to a distance less than d.