As per the given question,
Energy of the one dimensional simple harmonic oscillator "E_n=(n+\\dfrac{1}{2})\\hbar\\omega"
Now, we know that probability of the oscillator in nth excited state,
"P=e^{-\\beta E_n}"
Now,
probability of the oscillator being in the first excited state "P_1=e^{-\\beta E_1}"
probability of the oscillator being in the ground state "P_0=e^{-\\beta E_0}"
Hence, the required ratio "\\dfrac{P_1}{P_o}=\\dfrac{e^{-\\beta E_1}}{e^{-\\beta E_0}}=\\dfrac{e^{-\\beta (1+\\frac{1}{2})\\hbar\\omega}}{e^{-\\beta (0+\\frac{1}{2})\\hbar\\omega}}"
"=\\dfrac{e^{-\\beta (\\frac{3}{2})\\hbar\\omega}}{e^{-\\beta (\\frac{1}{2})\\hbar\\omega}}"
"=e^{-\\beta\\hbar\\omega(-\\frac{3}{2}+\\frac{1}{2})}"
"=e^{-\\beta \\hbar\\omega}"
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