Question

How do you write position and momentum in terms of the raising and lowering operators?

Answer 1

We can add these together or deduct them to discover the position and momentum operators regarding them. where ${\upsilon _x}$ is the vibrational quantum number for the $x$ measurement, and goes as $0$ , $1$ , $2$ .

Complete step by step answer:
In straight polynomial math (and its application to quantum mechanics), a raising or bringing down administrator (all things considered known as stepping stool operators) is an administrator that increments or diminishes the eigenvalue of another administrator.
These are known as the bringing down and raising operators, individually, for reasons that will before long get apparent. Dissimilar to $x$ and $p$ and the wide range of various operators we've worked with up until this point, the bringing down and raising operators are not Hermitian and don't represent any discernible amounts.
Disarray emerges on the grounds that the term stepping stool administrator is ordinarily used to depict an administrator that demonstrates to addition or decrement a quantum number portraying the condition of a framework. To change the condition of a molecule with the conception/destruction operators of $QFT$ requires the utilization of both a demolition administrator to eliminate a molecule from the underlying state and a creation administrator to add a molecule to the last state.
The expression "stepping stool administrator" is likewise at times utilized in science, with regards to the hypothesis of Lie algebras and specifically the relative Lie algebras, to portray the $su\left( 2 \right)$ subalgebras, from which the root framework and the most noteworthy weight modules can be developed by methods for the stepping stool operators. Specifically, the most noteworthy weight is destroyed by the raising operators; the remainder of the positive root space is obtained by consistently applying the bringing down operators (one bunch of stepping stool operators per subalgebra.

Note: Ladder operators (talked about in area $3$ of part $5$ in AIEP volume $173$ ) are - operators can be applied to turn or orbital precise momentum or their whole or resultant. The documentation as far as the stepping stool operators exhibits the total. Note that we didn't have to know the estimation of $\lambda l$ to make this inference.