Selection Rules (Time Dependent Perturbation Theory)

In summary, the selection rules for the matrix element of an operator $X$ state that it will be equal to zero, unless the difference in angular momentum and magnetic quantum numbers are equal to certain constants $\Delta l$ and $\Delta m$. For the position operator, these constants are $\pm 1$. This is in accordance with Shankar's specification.
  • #1
Diracobama2181
75
2
Homework Statement
Suppose that the system is a single spinless particle of mass $$M$$ and charge $$e$$ in a central Coulomb potential and the quantum numbers of the initial state are labeled $$|n, l, m>$$ in the usual way. It is subject to perturbation with magnetic field $$ B(t) = B_0e ^{−λt}$$ pointing in the x direction, which adds a term $$eL_xB(t)/(2Mc)$$ to the Hamiltonian. Find the quantum number selection rules for allowed transitions to a different state.
Relevant Equations
$$[L_x,X]=0$$
I suppose my question is, since X commutes for H, does this mean that the selection rules are $$<n',l',m'|X|n,l,m>=0$$ unless $$l'=l\pm 1$$ and $$m'=m\pm 1$$, as specified in Shankar?
 
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  • #2
No, that's not correct. The selection rules for the matrix element of any operator $X$ are given by $$\langle n',l',m'|X|n,l,m\rangle=0$$ unless $$l'-l=\Delta l$$ and $$m'-m=\Delta m$$ where $\Delta l$ and $\Delta m$ are constants depending on the particular operator $X$. In the case of the position operator, $\Delta l=\pm 1$ and $\Delta m=\pm 1$.
 

Related to Selection Rules (Time Dependent Perturbation Theory)

What is the concept of Selection Rules in Time Dependent Perturbation Theory?

Selection Rules refer to the conditions that determine whether a transition between two quantum states is allowed or forbidden. In Time Dependent Perturbation Theory, these rules are used to determine which transitions are possible between energy states in a system.

What are the three main types of Selection Rules in Time Dependent Perturbation Theory?

The three main types of Selection Rules are electric dipole selection rules, magnetic dipole selection rules, and electric quadrupole selection rules. These rules are based on the type of perturbation (electric or magnetic) and the angular momentum of the initial and final states.

How do electric dipole selection rules work?

Electric dipole selection rules are based on the principle that a transition between two states is allowed if the dipole moment of the system changes between the two states. This means that the initial and final states must have different dipole moments, and the transition can only occur if there is an external electric field present.

What is the significance of selection rules in spectroscopy?

Selection rules play a crucial role in interpreting spectroscopic data. By knowing which transitions are allowed or forbidden, we can determine the energy levels and electronic configurations of a system. This information is essential for understanding the properties and behavior of atoms, molecules, and other quantum systems.

Can selection rules be violated?

In general, selection rules are considered to be strict guidelines that cannot be violated. However, there are some cases where they can be violated, such as in high-energy processes or in certain symmetrical systems. These violations are rare and often require specific conditions to occur.

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