First Energy Level Meaning: What Does It Mean?

[omri3012]

Hallo.

my teacher wrote for the first energy level of a particle in a certain potential


$$\Delta X \Delta P \approx \frac{\hbar}{2}$$

exist.

is it a general result for all energy level or there is specific meaning for the first energy level?

 

[alxm]

Well $$\Delta X\Delta P \ge \hbar/2$$ always, per the uncertainty principle. For the lowest energy state of a harmonic oscillator, a gaussian wave-packet has the lowest ground-state energy, and for the ground state $$\Delta X\Delta P = \hbar/2$$.

The ground-state energy is not always related to the uncertainty principle like this; it depends on whether or how the terms are related to the energy levels described. With a harmonic oscillator potential, the energy levels are vibrational modes - so momentum and position are related in a fairly straightforward way.

It demonstrates the uncertainty principle and shows that bound particles cannot be perfectly stationary - and hence, that they must have a certain amount of kinetic energy even in their ground state, known as zero-point energy.

 

[naturale]

Hallo.

my teacher wrote for the first energy level of a particle in a certain potential


$$\Delta X \Delta P \approx \frac{\hbar}{2}$$

exist.

is it a general result for all energy level or there is specific meaning for the first energy level?

For the n-th energy level you have

$$\Delta X \Delta P = n \frac{\hbar}{2}$$

this is called Bohr-Sommerfeld quantization condition and say that the allowed orbits are those with an integer number of cycle. It is a periodicity condition.