Harmonic Oscillator Energy Levels

The energy levels ($E_n$) of a quantum mechanical one-dimensional simple harmonic oscillator (SHO) are quantized and given by the formula:

$ E_n = \left(n + \frac{1}{2}\right)h\nu $

where $n$ is the quantum number ($n = 0, 1, 2, ...$), $h$ is Planck's constant, and $\nu$ is the characteristic vibrational frequency.

Calculating Energy Spacing

The lowest energy level occurs when $n = 0$, known as the zero-point energy ($E_0$):

$ E_0 = \left(0 + \frac{1}{2}\right)h\nu = \frac{1}{2}h\nu $

We are given that the lowest energy is $E_0 = 300$ cm$^{-1}$.

$ \frac{1}{2}h\nu = 300 \text{ cm}^{-1} $

Therefore, the energy difference between adjacent levels, $h\nu$, is:

$ h\nu = 2 \times 300 \text{ cm}^{-1} = 600 \text{ cm}^{-1} $

This value, $h\nu$, represents the constant energy spacing between consecutive energy levels of the SHO.

Determining the Next Higher Energy Level

The question asks for the energy of the *next higher level*. This corresponds to the state where $n=1$.

The energy of the next level ($E_1$) can be found in two ways:

• Using the formula: $E_1 = (1 + \frac{1}{2})h\nu = \frac{3}{2}h\nu = \frac{3}{2} \times (600 \text{ cm}^{-1}) = 900 \text{ cm}^{-1}$.
• By adding the energy spacing to the lowest energy: $E_1 = E_0 + h\nu = 300 \text{ cm}^{-1} + 600 \text{ cm}^{-1} = 900 \text{ cm}^{-1}$.

Both methods yield the same result.

Conclusion

The energy of the next higher level is $900$ cm$^{-1}$.