Question

Describe formally what are the following for the automaton M0: set of states, initial state, final state, and transition function.

a. States: {Q}, Initial: Q, Final: Q, Transition: δ
b. States: {S}, Initial: S, Final: S, Transition: τ
c. States: {A, B, C}, Initial: A, Final: C, Transition: ψ
d. States: {X, Y, Z}, Initial: X, Final: Y, Transition: Φ

Answer 1

The allowed quantum transitions must follow selection rules, with the quantum number of the initial state being crucial for determining if a transition is allowed. For an initial state with quantum number l=1, the resulting orbital angular momentum is L=1.41ħ.

Step-by-step explanation:

The question requires understanding of quantum transitions, specifically concerning the rules that govern allowed transitions in quantum systems. When considering transitions between quantum states in a system, certain selection rules must be adhered to, which dictate whether a transition between two states is allowed or forbidden. In the case provided, we must look at the quantum number l of the initial state. An allowed transition typically involves a change in the quantum number (represented by Δl).

For a transition to be allowed, if the quantum number of the initial state is l = 0, the transition is not possible since Δ≠ 0. Similarly, if the quantum number is l = 2, 3, 4,... then the transition is also forbidden because Δl > 1 is not allowed. Hence, for an allowed transition, the quantum number of the initial state must be l = 1. The resultant orbital angular momentum (L) for the initial state can be calculated using the given relationship L = √(l(l + 1))ℏ = 1.41ℏ where ℏ is the reduced Planck constant.

Answer 2

The set of states, initial state, final state, and transition function for the automaton M0 are described.

Step-by-step explanation:

For automaton M0:

a. States: {Q}, Initial: Q, Final: Q, Transition: δ
b. States: {S}, Initial: S, Final: S, Transition: τ
c. States: {A, B, C}, Initial: A, Final: C, Transition: ψ
d. States: {X, Y, Z}, Initial: X, Final: Y, Transition: Φ

The set of states represents the possible states that the automaton can be in. The initial state represents the starting state of the automaton, while the final state represents the state at which the automaton accepts the input. The transition function represents the rules that govern how the automaton transitions from one state to another based on the input it receives.