The state of an ammonia molecule approximated by a two-state system as in the Feynman Lectures can be written [itex]|\psi\rangle = C_{1}|1\rangle + C_{2}|2\rangle[/itex], where the general solution for the coefficients is(adsbygoogle = window.adsbygoogle || []).push({});

$$

C_{1} = \frac{a}{2}e^{i(E_{0} - A)t/\hbar} + \frac{b}{2}e^{i(E_{0} + A)/\hbar}$$

and

$$

C_{2} = \frac{a}{2}e^{i(E_{0} - A)t/\hbar} - \frac{b}{2}e^{i(E_{0} + A)/\hbar}$$

By taking [itex]a = \sqrt{2}[/itex] and [itex]b = 0[/itex], we get a stationary state [itex]|I\rangle[/itex] with energy [itex]E_{0} - A[/itex]. By taking [itex]a = 0[/itex] and [itex]b= \sqrt{2}[/itex], we get a stationary state [itex]|II\rangle[/itex] with energy [itex]E_{0} + A[/itex]. For each of these states, we have [itex]\Delta E =0[/itex].

By taking [itex]a = 1[/itex] and [itex]b = 1[/itex], the molecule is in state [itex]|1\rangle[/itex] at [itex]t = 0[/itex] and in state [itex]|2\rangle[/itex] at [itex]t = \frac{\pi \hbar}{2A}[/itex]. The probability oscillates between states [itex]|1\rangle[/itex] and [itex]|2\rangle[/itex], and the system is in a constant superposition of states [itex]|I\rangle[/itex] and [itex]|II\rangle[/itex], so we have [itex]\Delta E = A[/itex] for this state.

Now, what about the other possible solutions? For example, what does the solution with [itex]a = \sqrt{1.5}[/itex] and [itex]b = \sqrt{0.5}[/itex] stand for? What to make of the fact that the probability to find the molecule in state [itex]|1\rangle[/itex] would never be 1 in this state? Is there a connection to be made with the time-energy uncertainty relation?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Solutions for ammonia as two-state system

Loading...

Similar Threads - Solutions ammonia state | Date |
---|---|

I Question about the solution to the Harmonic wave equation | Feb 3, 2018 |

I How to identify admissible functions in QM by simple math? | Jan 16, 2018 |

A Non-selfadjointness and solutions | Jan 15, 2018 |

I Solutions to TDSE from TISE | Oct 18, 2017 |

I Molecular Hamiltonian - Ammonia | Dec 6, 2016 |

**Physics Forums - The Fusion of Science and Community**