Question

An ammonia molecule consists of three hydrogen atoms and one nitrogen atom, bound together by their shared electrons. The nitrogen can be either to the left or to the right of a plane formed by the hydrogen. We can model the nitrogen as a particle with mass $m = 3m_Nm_H/(m_N + 3m_H)$ moving in the potential $V(x) = \frac{m\omega_0^2(x^2 - x_0^2)^2}{8x_0^2}$, where $x_0 = 38.15\text{pm}$ and $\omega_0 = 3.033 \times 10^{14}\text{rad/s}$. At $t = 0$ the particle has the wave function $\psi(x) = N \exp\left[-(x - x_0)^2 / (4\sigma_0^2)\right]$ where $\sigma_0 = 6.439\text{pm}$. Explain physically why the potential has a maximum at $x = 0$. Determine the expected value of the particle's kinetic and potential energy, $\langle T\rangle$ and $\langle V\rangle$. According to classical physics, would it have been enough for the nitrogen to end up to the left of the hydrogen atoms?

          An ammonia molecule consists of three hydrogen atoms and one nitrogen atom, bound together by
their shared electrons. The nitrogen can be either to the left or to the right of a plane formed by the
hydrogen. We can model the nitrogen as a particle with mass $m = 3m_Nm_H/(m_N + 3m_H)$ moving in
the potential
$V(x) = \frac{m\omega_0^2(x^2 - x_0^2)^2}{8x_0^2}$,
where $x_0 = 38.15\text{pm}$ and $\omega_0 = 3.033 \times 10^{14}\text{rad/s}$. At $t = 0$ the particle has the wave function
$\psi(x) = N \exp\left[-(x - x_0)^2 / (4\sigma_0^2)\right]$
where $\sigma_0 = 6.439\text{pm}$.
Explain physically why the potential has a maximum at $x = 0$. Determine the expected value of
the particle's kinetic and potential energy, $\langle T\rangle$ and $\langle V\rangle$. According to classical physics, would it have
been enough for the nitrogen to end up to the left of the hydrogen atoms?

An ammonia molecule consists of three hydrogen atoms and one nitrogen atom, bound together by
their shared electrons. The nitrogen can be either to the left or to the right of a plane formed by the
hydrogen. We can model the nitrogen as a particle with mass m = 3mNmH/(mN + 3mH) moving in
the potential
V(x) = (mω0^2(x^2 - x0^2)^2)/(8x0^2),
where x0 = 38.15pm and ω0 = 3.033 × 10^14rad/s. At t = 0 the particle has the wave function
ψ(x) = N [-(x - x0)^2 / (4σ0^2)]
where σ0 = 6.439pm.
Explain physically why the potential has a maximum at x = 0. Determine the expected value of
the particle's kinetic and potential energy, ⟨ T⟩ and ⟨ V⟩. According to classical physics, would it have
been enough for the nitrogen to end up to the left of the hydrogen atoms?

Added by Charles R.

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