Importance of raising/lowering/ladder operators

Question

While learning quantum mechanics I repetitively encountered the concepts of raising and lowering operators. Firstly in the harmonic oscillator $a_\pm \ $and now in angular momentum I am introduced to $J_\pm$ raising and lowering operators. Why this idea of these operators are of so importance in quantum mechanics? As they are abstract, what do they mean in physical approach? I mean can't we find some analytical approach instead of using them?

Answer 1

That is just the thing about quantum mechanics. It is based on the fundamental principle that interactions involve the exchange of quanta. As a result of this principle each particle field taking part in the interaction contributes or receives a quantum to or from the interaction. It was this principle that help Max Planck to solve the problem of black body radiation. So when we model such interactions in our theories, we find it convenient to use such raising and lowering operators. Theories that are based on this principle (using quantum field theory for instance) have been found to be quite successful when their predictions were compared with experimental result.

If by "some analytical approach" you mean an approach not based on operators, then there is indeed a different formalism. It is called the Moyal formalism and is based on operations among distribution functions on phase space instead of operators. It is not as popular as the operator approach.

Answer 2

The construction of raising and lowering operator is very natural when there is an underlying Lie algebra. Thus yes there's some systematic way of constructing them. In fact, if the Lie algebra is of the semi-simple type, then the construction of representations is completely based on raising and lowering operators.

If the Lie algebra is not semi-simple, one can use Levi's theorem which very loosely states that any Lie algebra has a semi-simple part and "another" part (called a radical) that contains operators (tensor operators) which generalize the idea of raising and lowering operators.

In your examples, the operators $\hat a,\hat a^\dagger$ and $\hat 1$ span the Heisenberg-Weyl Lie algebra $\mathfrak{hw}(1)$. The angular momentum operators $\hat L_x,\hat L_y$ and $\hat L_z$ span the Lie algebra $\mathfrak{su}(2)$. (Actually the raising and lowering operators are often complex combinations of the observables in your model, so you need to first consider these operators over the complex field and then possibly go back to the reals but that's a technical matter. )

All the $\mathfrak{su}(n)$ algebras are semi-simple and the representations are constructed using raising and lowering operators. With $i,j$ running from $1$ to $n$, the operators $\hat C_{ij}=a_i^\dagger a_j$ with $i<j$ can be taken as raising operators, with their hermitian conjugates $\hat C_{ji}= a^\dagger_j a_i$ the lowering operators, acting on basis states where $\hat{C}_{ii}-\hat{C}_{jj}=a_i^\dagger a_i-a_j^\dagger a_j$ is diagonal.

Likewise representations of the symplectic algebras $\mathfrak{sp}(2n,\mathbb R)$ are constructed by first considering the $\hat C_{ij}$ of $\mathfrak{u}(n)$ and adding $\hat A_{ij}=a^\dagger_ia^\dagger_j$ and $\hat B_{ij}= a_i a_j$. The $\hat A_{ij}$'s and $\hat B_{ij}$'s are raising and lowering respectively, in addition to the raising and lowering operators in $\mathfrak{u}(n)$.

The Lie algebra $\mathfrak{e}(2)$ is not semi-simple. It is spanned by $\hat L_z,\hat p_x$ and $\hat p_y$. Yet, it's possible to define $\hat p_\pm = p_x\pm i p_y$ so that $\hat p_\pm$ act by "raising" and "lowering" action on the eigenstates of $\hat L_z$ much in the way that $\hat L_\pm$ act by raising and lowering.