# QM: Operator algebra and inner products

• Niles
In summary, the teacher's book says that z* should be in front of the inner product, while the book used by Niles says that z* should be in the back.
Niles

## Homework Statement

We have a hermitian operator Q and a complex number z. I have an inner product given by:

<f | (zQ) g>,

where f and g are two arbitrary functions. Now taking the hermitian conjugate of a complex number is just taking the complex conjugate of that number. So taking the hermitian conjugate of the above gives me:

<(zQ)H f | g> = <z*QH f | g> = z<QH f | g> = z<Qf | g>

According to my book, this is wrong, because I should end up with z* in front of the inner product, and not z. Where is my reasoning wrong?

It also depends on the book used, some authors prefer to define

$$\langle f | z g \rangle = z \langle f | g \rangle, \langle z f | g \rangle = z^* \langle f | g \rangle$$

while some use

$$\langle f | z g \rangle = z^* \langle f | g \rangle, \langle z f | g \rangle = z \langle f | g \rangle$$

A teacher of mine (mathematician who worked closely with physicists) once said that one is typically used by mathematicians while the other is often used by physicists. You are free to choose which convention you use, as long as you use it consistently.

My book uses the first one you mentioned. But I am using the same convention, so this is what troubles me.

Can you spot my error? The place where my book is different is when taking the hermitian conjugate - there the author does not complex conjugate the number z (actually, he does not even assume it is complex.)

I am using Griffiths Introduction to QM.

I think you are right, in fact you can take out the z right away:
<f | z Q g> = z <f | Q g> = z <Q f | g>
by the property I mentioned and the hermiticity of Q, respectively.

Niles said:
Griffiths Introduction to QM.

I have that book in front of me right now... can you give me a page number?

Last edited:
It is problem 3.4, part B on page 110, chapter 3.

But I think you solved my mystery. Because this just tells us that the complex number z has to be a real number, which is the answer to that problem.

Since you are sitting with the book in front of you, I hope it is OK I ask another question on this topic. It is on problem 3.5 part C on the same page. This is my approach:

<f | (QR) g> = <(QR)Hf | g> = <QHRH f | g>,

which is not what I have to show. But isn't the hermitian conjugate a "linear operator" so to speak? That is what I had in mind when solving it.

Thanks in advance, I really appreciate it.

For matrices you have the property that
(A B)T = BT AT,
while complex conjugation doesn't change the order:
(A B)* = A* B*

Therefore, the Hermitian conjugate works like the transpose as far as ordering of matrices is concerned:
(A B C ...)H = ... CH BH AH

You're welcome, glad I could help.

## 1. What is the purpose of operator algebra in quantum mechanics?

Operator algebra is a mathematical framework used to describe the behavior of quantum systems. It allows us to manipulate and calculate properties of quantum operators, which represent physical observables such as position, momentum, and energy.

## 2. How do inner products relate to quantum mechanics?

In quantum mechanics, inner products are used to calculate the probability of a quantum system transitioning from one state to another. They also play a crucial role in determining the expectation values of physical observables.

## 3. Can you explain the difference between Hermitian and unitary operators?

Hermitian operators are those that are equal to their own adjoint (conjugate transpose), while unitary operators are those that preserve the norm of a vector. In quantum mechanics, Hermitian operators represent physical observables, while unitary operators represent transformations between quantum states.

## 4. How are operator algebras used in quantum computing?

In quantum computing, operator algebras are used to manipulate and control the behavior of quantum bits (qubits). This allows for complex calculations and operations to be performed in a quantum system.

## 5. What are some common examples of inner product spaces in quantum mechanics?

Some common examples of inner product spaces in quantum mechanics include the space of wavefunctions, the space of quantum states, and the space of operators. These spaces allow for mathematical representations of physical quantities and their relationships within a quantum system.

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