# Complex operations, bra-ket notation, confusion

• demoncore
In summary, the conversation discusses the introduction of bra-ket notation in quantum mechanics and its use in the Stern-Gerlach experiment. They also mention the experimental results and the concept of coefficients being complex and having an amplitude and phase. The reasoning behind choosing one coefficient to be real and positive without loss of generality is also discussed.
demoncore
Sorry for disregarding the template; I'm not really working out a homework problem as much as just trying to follow the reasoning in the text. I'm working through the first chapter of Quantum Mechanics, McIntyre, and I'm a little bit confused by the following.

The text introduces bra-ket notation in the context of the Stern-Gerlach experiment, deriving some results having to do with the spin measured along various axes and relating it to the state vector. (Please read the _ sign as a subscript)

They write the 'general form' of the S_x state kets in terms of the S-z bases kets:

|+>_x = a |+> + b |->
|->_x = c |+> + d |->,

They also have the following experimental results:

|<+|+>|^2 = |<-|+>|^2 = |<+|->|^2 = |<-|->^2 = 0.5 (all S_x state vectors)

expanding and equating they find that |a|^2 = |b|^2 = |c|^2 = |d|^2 = 0.5

This is the paragraph that confuses me--I think it might be my unfamiliarity with complex operations:
Because each coefficient is complex, each has an amplitude and phase. However, the overall phase of a quantum state vector is not physically meaningful. Only the relative phase between different components of the state vector is physically measurable. Hence, we are free to choose one coefficient of each vector to be real and positive without any loss of generality. This allows us to write the desired states as:

|+>_x = (1/√2) [ |+> + e^(iα) |->]
|->_x = (1/√2) [ |+> + e^(iβ) |->]

I am a little confused by the reasoning here. I suppose |a|^2 = (1/√2) => a = (1/√2) * e^(iθ); the absolute phase of the state vector doesn't matter, so we are free to take θ= 0 for one coefficient of each vector?

I suppose [since] the absolute phase of the state vector doesn't matter, so we are free to take θ= 0 for one coefficient of each vector?
Where you are free to pick a relationship, pick one that makes the math easy.

## 1. What are complex operations in quantum mechanics?

Complex operations in quantum mechanics refer to mathematical operations that involve complex numbers. In quantum mechanics, physical quantities such as energy and position are represented by complex numbers, and operations involving these quantities are performed using complex numbers.

## 2. What is bra-ket notation used for in quantum mechanics?

Bra-ket notation, also known as Dirac notation, is used in quantum mechanics to represent quantum states and operators. The notation uses the symbols bra and ket to denote the states and operators, which are represented by bra and ket vectors respectively.

## 3. Why is confusion common when learning about complex operations and bra-ket notation?

Complex operations and bra-ket notation can be confusing because they involve abstract mathematical concepts that are not commonly encountered in everyday life. Additionally, quantum mechanics itself is a complex and counterintuitive theory, which can make it difficult to grasp at first.

## 4. How do complex operations and bra-ket notation relate to each other in quantum mechanics?

Complex operations and bra-ket notation are closely related in quantum mechanics. Bra-ket notation is used to represent the mathematical operations that are involved in quantum mechanics, and these operations often involve complex numbers. The notation also helps to simplify and streamline the complex calculations in quantum mechanics.

## 5. Are there any tips for understanding complex operations and bra-ket notation in quantum mechanics?

One tip for understanding complex operations and bra-ket notation in quantum mechanics is to practice and get familiar with the basic principles and concepts. Building a strong foundation in quantum mechanics can help with understanding the more complex operations and notation. Additionally, seeking help from a knowledgeable instructor or studying from reliable sources can also be beneficial.

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