# Solving Creation Operator Equation: Find Error in Calculation

• Luke Tan
In summary, the conversation discusses the equation and adjoint operation involving an unknown constant. The incorrect use of the commutator is identified and the correct expression is derived. The final result is that the constant is equal to the square root of n + 1.
Luke Tan
Homework Statement
Show that ##a^\dagger\lvert n \rangle = \sqrt{n+1}\lvert n+1 \rangle##
(From Shankar, Principles of Quantum Mechanics, Chapter 8 - The Harmonic Oscillator)
Relevant Equations
##\hat{H}+\frac{1}{2}=aa^\dagger=a^\dagger a##
##\hat{H}\lvert n \rangle = (n+\frac{1}{2})\lvert n \rangle##
I have written the equation, with an unknown constant
$$a^\dagger \lvert n\rangle = C_{n+1}\lvert n+1 \rangle$$
I then take the adjoint to get
$$\langle n \rvert a = \langle n+1 \rvert C_{n+1}^\text{*}$$
I then multiply them to get
$$\langle n \rvert aa^\dagger \lvert n \rangle = \langle n+1 \rvert n+1 \rangle |C_{n+1}|^2$$
On the left hand side, since ##aa^\dagger = \hat{H}-\frac{1}{2}##, the expression just simplifies to ##\langle n \rvert n \lvert n \rangle##. On the right hand side, since ##\lvert n+1 \rangle## is a normalized state, it just simplifies to ##|C_{n+1}|^2##. Thus, we arrive at
$$\langle n \rvert n \lvert n \rangle = |C_{n+1}|^2$$
$$n = |C_{n+1}|^2$$
$$C_{n+1}=\sqrt{n}$$.
Thus,
$$a^\dagger \lvert n \rangle = \sqrt{n} \lvert n+1 \rangle$$
Which is wrong.

I can't see where i went wrong. Can someone help?

First error I see is where you wrote: $aa^\dagger = a^\dagger a$ their commutator is not 0.

jambaugh said:
First error I see is where you wrote: $aa^\dagger = a^\dagger a$ their commutator is not 0.

Would it make any difference?
From my equation

$$\langle n \rvert aa^\dagger \lvert n \rangle = \langle n+1 \rvert n+1 \rangle |C_{n+1}|^2$$

Could i take the adjoint to get
$$\langle n \rvert a^\dagger a \lvert n \rangle = \langle n+1 \rvert n+1 \rangle |C_{n+1}|^2$$

which would result in the same expression?

The number operator is self-adjoint (conjugate transpose). Yes it makes a big difference!

Luke Tan said:
Could i take the adjoint to get
That is wrong. The adjoint of ##a a^\dagger## is ##aa^\dagger##, not ##a^\dagger a##. Note that ##(ab)^\dagger = b^\dagger a^\dagger## for general a and b.

Oh i got the answer

We start from the original equation
$$a^\dagger \lvert n\rangle = C_{n+1}\lvert n+1 \rangle$$

We then take the adjoint to get
$$\langle n \rvert a = \langle n+1 \rvert C_{n+1}^\text{*}$$

Which we then combine to arrive at
$$\langle n \rvert aa^\dagger \lvert n \rangle = \langle n+1 \rvert n+1 \rangle |C_{n+1}|^2$$

Since we know the commutator ##[a,a^\dagger]=aa^\dagger-a^\dagger a=1##, we can then derive ##aa^\dagger = 1 + a^\dagger a##. We then substitute this to get

$$\langle n \rvert 1 + a^\dagger a \lvert n \rangle = \langle n+1 \rvert n+1 \rangle |C_{n+1}|^2$$

Which then simplifies to

$$\langle n \rvert \hat{H}+\frac{1}{2} \lvert n \rangle = |C_{n+1}|^2$$

Which evaluates to ##\langle n \rvert n+1 \lvert n \rangle = |C_{n+1}|^2##

Thus, we get ##|C_{n+1}|=\sqrt{n+1}##

Thanks everyone for the help!

## What is a creation operator equation?

A creation operator equation is an equation used in quantum mechanics to describe the creation of particles in a system. It involves operators, which are mathematical tools used to manipulate quantum states.

## What is the purpose of solving a creation operator equation?

The purpose of solving a creation operator equation is to accurately determine the probability of finding a certain number of particles in a given quantum state. This information is important in understanding the behavior of a system and making predictions about its future state.

## What is the process for solving a creation operator equation?

The process for solving a creation operator equation involves using mathematical tools such as commutators and eigenvalue equations to manipulate the operators and simplify the equation. This can be a complex process and may require the use of advanced mathematical techniques.

## What are some common errors that can occur when solving a creation operator equation?

Some common errors that can occur when solving a creation operator equation include incorrect use of operators, incorrect commutator algebra, and mistakes in simplification. It is important to carefully check each step of the solution to avoid these errors.

## What are some tips for avoiding errors when solving a creation operator equation?

To avoid errors when solving a creation operator equation, it is important to have a strong understanding of quantum mechanics and the mathematical tools involved. It is also helpful to double-check each step of the solution and to seek assistance from a colleague or mentor if needed.

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