The raising and lowering operators change the value of m by...

The raising and lowering operators change the value of $m$ by one unit: $$ L_{\pm} f_{l}^{m}=\left(A_{l}^{m}\right) f_{l}^{m \pm 1} $$ where $A_{l}^{m}$ is some constant. Question: What is $A_{l}^{m}$, if the eigenfunctions are to be normalized? Hint: First show that $L_{\mp}$ is the hermitian conjugate of $L_{\pm}$ (since $L_{x}$ and $L_{y}$ are observables, you may assume they are hermitian $\ldots$ but prove it if you like); then use Equation 4.112. Answer: $$ A_{l}^{m}=\hbar \sqrt{l(l+1)-m(m \pm 1)}=\hbar \sqrt{(l \mp m)(l \pm m+1)} $$ Note what happens at the top and bottom of the ladder (i.e., when you apply $L_{+}$ to $f_{l}^{l}$ or $L_{-}$ to $\left.f_{l}^{-l}\right)$.

The raising and lowering operators change the value of $m$ by one unit:
$$
L_{\pm} f_{l}^{m}=\left(A_{l}^{m}\right) f_{l}^{m \pm 1}
$$
where $A_{l}^{m}$ is some constant. Question: What is $A_{l}^{m}$, if the eigenfunctions are to be normalized? Hint: First show that $L_{\mp}$ is the hermitian conjugate of $L_{\pm}$ (since $L_{x}$ and $L_{y}$ are observables, you may assume they are hermitian $\ldots$ but prove it if you like); then use Equation 4.112. Answer:
$$
A_{l}^{m}=\hbar \sqrt{l(l+1)-m(m \pm 1)}=\hbar \sqrt{(l \mp m)(l \pm m+1)}
$$
Note what happens at the top and bottom of the ladder (i.e., when you apply $L_{+}$ to $f_{l}^{l}$ or $L_{-}$ to $\left.f_{l}^{-l}\right)$.

Key Concepts

Normalization of Eigenstates

Normalization of eigenstates ensures that the probability interpretation of quantum mechanics is maintained, meaning the total probability of finding a system in any state sums to one. In this context, finding the correct constant (A??) when applying ladder operators ensures that the resulting eigenstates remain normalized.

Ladder Structure and Boundary Conditions

The ladder structure inherent in the use of raising and lowering operators reflects the discrete nature of the magnetic quantum number. The top and bottom of the ladder, where the application of L? or L? yields zero, correspond to the maximum and minimum allowed values of m, respectively, hence enforcing the quantization of angular momentum.

Hermitian Conjugation

Hermitian conjugation is a process applied to operators in quantum mechanics to yield their adjoint. Observables, such as the angular momentum components, are represented by Hermitian operators. The fact that L? is the Hermitian conjugate of L? allows one to derive important symmetry and normalization conditions for the eigenstates and the constants appearing in the ladder operator expressions.

Angular Momentum Operators

Angular momentum operators in quantum mechanics represent the components of angular momentum, which is fundamental to rotational symmetry in physical systems. They obey specific commutation relations that determine the properties of the system and allow the derivation of quantized angular momentum values.

Raising and Lowering (Ladder) Operators

The raising (L?) and lowering (L?) operators, often called ladder operators, are used to change the magnetic quantum number (m) of the angular momentum eigenstates by one unit. They are crucial for constructing the full set of eigenstates of the angular momentum operator and exploring the structure of the angular momentum spectrum.

Transcript

00:01 Hello everyone so we are doing the problem 4 .18 of grits here we have to prove the following equation given in the question for that first of all we have to prove that l plus or minus is actually the hermine hermitian conjugate of l minus plus for this we will evaluate this equation l plus minus g which is given by f l lxg plus minus i f, ly, g.

00:51 So this is the definition of l plus or minus.

00:54 We have just applied the definition l plus minus l plus minus l.

00:58 L plus minus i, l .y, which is equals to.

01:03 Now since lx and ly are hermission, so we can write it as lxfg plus minus i, ly, fg.

01:24 And hence, we can write this as lx plus minus.

01:33 But here, since there is a minus sign here, hence you can change the minus signs.

01:43 Since i, if we take it inside, this will give back a minus sign here and it will change the sign here.

01:54 So lx minus plus i, ly, the whole of fg.

02:07 Notice that the i, if it goes inside, it gives back a minus sign because the first operator here is a star inside an integral.

02:17 So if i will go inside, it will give back a minus sign.

02:22 And hence, it will become minus plus.

02:25 So this is nothing but our l of, minus plus fg.

02:34 This means that the hermitian conjugate of l plus minus is actually l minus plus...

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