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Quantum Mechanics Tutorials Study of a two-state system : the ammonia molecule Study of a two-state system : the ammonia molecule We first consider the ammonia molecule in the absence of any external perturbation. The nitrogen atom can be above or below the plane P defined by the 3 hydrogen atoms. This defines 2 possible states of the molecule. We will denote by direction with respect to the plane P and by the state where the nitrogen atom points towards the +z the state where the nitrogen atom points towards the - z direction. The ammonia molecule always switches from the state 1) Represent the molecule in the 2 states and to the state . . In the following we will assume the these two states form an orthonormal basis set of the accessible states of the molecule. If is the state of the system at time t, then the evolution of with t is given by the time- dependent Schrödinger equation: , where is the hamiltonian of the system. 2) Are the states and eigenstates of the system ? (justify your answer). 3) At time t the system is in the state where and are two complex numbers that depend on time. What is their physical meaning ? 4) The matrix representation of the hamiltonian in the basis of the states and can be written as follows: . Show that and are real numbers and that . Prove that the time-dependent Schrödinger equation is equivalent to: 10 Quantum Mechanics Tutorials . 5) If and and switching from are not eigenstates of the system, what are the consequences on the coefficients ? Prove that if the molecule can switch from the state to ( , then is also possible. 6) For symmetry reasons and is denoted to the state = = . In addition, the coupling term ). What are the normalized eigenvectors What are the corresponding energies and ? The state molecule, that is the state associated to the lowest energy 7) If, at time t=0 the molecule is in the state and is assumed to be real of the hamiltonian ? is chosen to be the ground state of the . , what is the probability that it is found in the state at time t ? Hint: solve the time-dependent Schrödinger equation in the basis of the states ( ). 8) We now perturb the system with a static electric field permanent dipole moment and ( ). The molecule has a . Its interaction energy with the electric field equals (in classical mechanics): . When the nitrogen atom is below the plane P, ( Explain why, in quantum mechanics, ). and . Give the matrix representation of the hamiltonian in the presence of the electric field. What are the possible energies for the molecule ? We denote the ground state energy and its associated eigenvector. 9) For analysis purposes, we will assume that the geometry of the molecule is not affected by the electric field, which means that the coupling term when the electric field is strong ( can be considered as a constant. Show that, ) and the molecule is in the ground state 11 , the probability Quantum Mechanics Tutorials of finding the nitrogen atom below the plane P is much larger than the probability of finding it above the plane. What are those probabilities equal to in the absence of the electric field ? 12 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !