# Quantum Mechanics - Induction Method

• izzmach
It's because the exact words are important and it's also very hard to figure out what someone is talking about if you don't know the relevant equations.
izzmach
Let a be a lowering operator and a† be a raising operator.

Prove that a((a†)^n) = n (a†)^(n-1)

Professor suggested to use induction method with formula:

((a†)(a) + [a,a†]) (a†)^(n-1)

But before start applying induction method, I would like to know where the given formula comes from. Someone please explain it briefly? Thank you.

If ##a## is a lowering operator and ##a^\dagger##
They are also known as "ladder" operators ... because ##a## means "go down one step" and ##a^\dagger## means "go up one step".
##(a^\dagger)^n## means "go up n steps"... and so on.
##a(a^\dagger)## means "go up one step then go down one" (operators work right-to-left).

Therefore the formula ##a(a^\dagger)^n = n(a^\dagger)^{(n-1)}## means ...

izzmach said:
Let a be a lowering operator and a† be a raising operator.
Prove that a((a†)^n) = n (a†)^(n-1)
[...]
You also have a https://www.physicsforums.com/threads/quantum-mechanics-lowering-operator.820823/[/url , so this discussion should probably continue there.

In any case, I think you have not stated the problem question accurately. (In general, it is "commutation by ##A##" which can be interpreted as "differentiation by ##A^\dagger##". Your formula only applies if the expression acts on a vacuum state ##|0\rangle## which is annihilated by ##A##.)

There is a reason why the homework guidelines emphasize that you must state the question exactly as given, and also write out relevant equations (in this case, the commutation relation between ##A## and ##A^\dagger##).

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## What is the Induction Method in Quantum Mechanics?

The Induction Method is a mathematical approach used in quantum mechanics to calculate the probability of a particle's position or momentum at a given time. It involves using the Schrödinger equation to determine the wave function of a particle, and then applying operators to the wave function to calculate various physical properties.

## How does the Induction Method differ from other methods in Quantum Mechanics?

The Induction Method differs from other methods in quantum mechanics, such as the Perturbation Theory or the Variational Method, in that it is a non-perturbative approach. This means that it does not rely on small perturbations or approximations, but instead calculates the exact solution to the Schrödinger equation.

## What are the applications of the Induction Method in Quantum Mechanics?

The Induction Method has many applications in quantum mechanics, including calculating the energy levels and wave functions of atoms and molecules, predicting the behavior of particles in a magnetic field, and studying the properties of quantum systems in general. It is also used in fields such as quantum chemistry, materials science, and quantum computing.

## What are the limitations of the Induction Method in Quantum Mechanics?

While the Induction Method is a powerful tool in quantum mechanics, it does have some limitations. It can only be applied to systems with a finite number of particles, and it becomes increasingly complex and computationally intensive for larger systems. It also does not take into account the effects of relativity and cannot be used for systems with strong interactions.

## What is the future of the Induction Method in Quantum Mechanics?

The Induction Method is constantly being improved and refined, and it will continue to play a crucial role in advancing our understanding of quantum mechanics. With the development of more powerful computers and new mathematical techniques, it is possible that the Induction Method may be able to tackle more complex systems and provide even more accurate predictions in the future.

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