Hydrogen molecule as a harmonic oscillator

1. The problem statement, all variables and given/known data
The covalent bond between the two protons in the H2 molecule can be modelled by a harmonic spring with a spring constant k = 1100 N/m.

a. What is the zero point energy of the H2 molecule as a harmonic oscillator?

b. What is the energy of the first excited state?

[tex]m_{p}=1.6726*10^-27 kg[/tex]

[tex]k=1100N/m[/tex]

2. Relevant equations
[tex]m=\frac{m_{p}}{2}[/tex]

[tex]\omega=\sqrt{\frac{k}{m}[/tex]

[tex]E=(n+\frac{1}{2})\hbar\omega[/tex]

3. The attempt at a solution

[tex]\omega=\sqrt{\frac{1200}{8.363*10^-28}[/tex]

a. [tex]E_{0}=\frac{1}{2}\hbar\omega=6.047J*10^-20J[/tex] this is CORRECT

b. ??[tex]E_{1}=\frac{3}{2}\hbar\omega=1.814*10^-19J[/tex] ?? this is INCORRECT

i don't understand why b. is incorrect but a. is correct. i thought the only difference would be the number for n. what am i doing wrong? can anyone help please?

 

thanks. i don't use that coding often. it should all be fixed now