How to illustrate the difference between quantum mechanics and ordinary mechanics through one-dimensional harmonic oscillators

Introduction of one-dimensional harmonic oscillator:

The so-called harmonic oscillation, in kinematics, is simple harmonic vibration. A coefficient of strength of k of a lightweight spring is fixed at one end, the other end of the solid can be free to move the mass of the object m, constitutes a spring oscillator the oscillator is in a position (i.e., the equilibrium position) near the reciprocating motion. In this form of vibration, the magnitude of the force on the object is always proportional to its distance from the equilibrium position, and the direction of the force is always pointing to the equilibrium position. This case is known as a one-dimensional harmonic oscillator.

The difference and connection between one-dimensional harmonic oscillator in classical mechanics and quantum mechanics

Classical harmonic oscillator and quantum harmonic oscillator have the essential difference, but the difference between them is not insurmountable, we will discuss and compare them one by one in terms of the energy level and the wavefunction, etc.

1.

1 . Difference between classical and quantum harmonic resonators

① Characteristics of energy values

By the kinetic and potential energies of classical harmonic resonators :?

It can be seen that the energy fetch of classical harmonic oscillator is continuous.

From the energy of the quantum harmonic oscillator : ?

It is known that : the energy of the quantum harmonic oscillator takes discrete values, i.e., quantized, and the energy levels are equidistant, with a spacing

of. The fact that the energy takes discrete values is an important manifestation of the quantum feature that microscopic particles have wave-particle duality.

Because the energy level spacing is equally spaced, the leap between energy levels only occurs between adjacent energy levels, i.e., the leap can only take place level by level, so each leap occurs with the same frequency radiation. The experimentally measured energy spectrum has only one spectral line. The energy spectrum has only one spectral line, which means that the leaps can only take place one at a time, so that all the leaps emit radiation at the same frequency. The experimentally measured energy spectrum has only one spectral line.

② Zero-point energy discussion

By and by equation (4.1.3), the quantum harmonic oscillator does not have zero energy in the ground state. That is, when n = 0, the time is called the zero-point energy. This is completely different from the classical harmonic oscillator, which once again shows that microscopic particles have wave-particle duality, there is no absolute "stationary" wave.

The zero-point energy of a quantum harmonic oscillator can also be estimated

directly using uncertainty relations in quantum mechanics. Using the uncertainty relation between coordinates and momentum: ?

The uncertainty in the energy of the harmonic oscillator is:

The value that makes taking the minimum can be calculated by the extreme value condition:

. We find that,

Therefore the zero energy of the harmonic oscillator:

It can be seen that the harmonic oscillator ground state is the smallest uncertainty state of the harmonic oscillator problem, which is determined by its quantum nature.

3) The wave function of the harmonic oscillator

In quantum mechanics the wave function itself is meaningless, but the square of the absolute value of the wave function: is proportional to the chance of the

particle appearing at a point in space.

First we discuss the positional odds, the potential penetration, using the ground state as an example. For the ground state of a quantum harmonic oscillator: