If you try to increase the energy of a quantum harmonics oscillator by adding an amount of energy $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{h}\sqrt{{\mathbf{k}}_{\mathbf{s}}\mathbf{/}\mathbf{m}}$, the energy doesn’t increase. Why not?

The expression for the energy of photon is given by,

${\mathrm{E}}_{\mathrm{photon}}=\frac{\mathrm{hc}}{{\mathrm{\lambda }}_{\mathrm{light}}}$

Here ${\mathrm{E}}_{\mathrm{photon}}$is the energy of the photon,$\mathrm{c}$ is the speed of the light, is the Planck’s constant,

The value of $\mathrm{c}$in vacuum is $3.0\times {10}^8\mathrm{m}/\mathrm{s}$.

The expression for the energy of the oscillator is given by,

${\mathrm{E}}_{\mathrm{N}}={\mathrm{Nh\omega }}_0+{\mathrm{E}}_0$...... (i)

Here $\mathrm{N}=0,1,2,\mathrm{....}$

The angular frequency is given by,

${\mathrm{\omega }}_0=\sqrt{\frac{{\mathrm{k}}_{\mathrm{s}}}{\mathrm{m}}}$

Here ${\mathrm{\omega }}_0$is the angular frequency, ${\mathrm{k}}_{\mathrm{s}}$is the spring constant, $\mathrm{m}$is the mass.

The ground state energy of the harmonic oscillator is given by,

${\mathrm{E}}_0=\frac{1}{2}{\mathrm{h\omega }}_0$ ....... (ii)

Substitute $\frac{1}{2}{\mathrm{h\omega }}_0$for ${\mathrm{E}}_0$into the equation (i)

$\begin{array}{c}{\mathrm{E}}_{\mathrm{N}}={\mathrm{Nh\omega }}_0+\frac{1}{2}{\mathrm{h\omega }}_0\\ =\left(\mathrm{N}+\frac{1}{2}\right){\mathrm{h\omega }}_0\end{array}$

Now if we try to increase the energy by $\frac{1}{2}\mathrm{h}\sqrt{\frac{{\mathrm{k}}_{\mathrm{s}}}{\mathrm{m}}}$$\left(\mathrm{or}\frac{1}{2}{\mathrm{h\omega }}_0\right)$then the new energy of the oscillator is,

$\begin{array}{c}{\mathrm{E}}_{\mathrm{N}}={\mathrm{Nh\omega }}_0+\frac{1}{2}{\mathrm{h\omega }}_0+\frac{1}{2}{\mathrm{h\omega }}_0\\ ={\mathrm{Nh\omega }}_0+{\mathrm{h\omega }}_0\\ =(\mathrm{N}+1){\mathrm{h\omega }}_0\end{array}$

But the vibration energy levels of harmonic oscillator are quantized in the fashion of $\left(\mathrm{N}+\frac{1}{2}\right){\mathrm{h\omega }}_0$, So we can’t increase the energy by $\frac{1}{2}\mathrm{h}\sqrt{\frac{{\mathrm{k}}_{\mathrm{s}}}{\mathrm{m}}}$$\left(\mathrm{or}\frac{1}{2}{\mathrm{h\omega }}_0\right)$ .