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jimmycricket
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Are all quantum states represented by normal vectors?
Major slip of the brain there, I should have read my post properly. What I meant to ask is: Are all quantum states represented by unit vectors in some complex vector space?Nugatory said:Every quantum state is represented by a vector in an abstract vector space.
What exactly do you mean by "normal"?
Is this what is referred to as a global phase factor?vanhees71 said:No, pure states are never represented by unit vectors but by unit rays (i.e. by unit vectors modulo an arbitrary phase factor)
So is the direction of a vector component the defining feature for a pure state? This makes sense because two rays with the same direction but different magnitude would give the same expectation value with respect to the same observable since they are identical unit vectors once normalized. Right?Nugatory said:I would say that they are represented by rays, where a ray is a set of all vectors that are a complex constant multiple of one another (same "direction", different magnitudes).
I thought that pure states corresponded to points on the bloch sphere which are defined by unit vectors aren't they?vanhees71 said:It is an important point to make clear that pure states are represented as rays not as vectors in Hilbert space.
jimmycricket said:So what exactly are they there for or why do they appear?
jimmycricket said:I must admit I've had a difficult time getting to grips with these global phase factors. I am currently writing a project on the basics of quantum information and quantum computing and they crop up everywhere. I understand that global phase factors are essentially irrelevant since they do not affect the expectation values for any observables. My proffesor has brushed over the toic when I hav mentioned it. So what exactly are they there for or why do they appear?
Quantum states as normal vectors are mathematical representations of the state of a quantum system. They describe the probability of finding a particle in a particular state, and can be used to calculate the outcome of measurements on the system.
Quantum states are represented as normal vectors in a mathematical space known as Hilbert space. Each possible state of a quantum system is represented by a different vector in this space, with the length and direction of the vector corresponding to the probability of finding the system in that state.
Using normal vectors to represent quantum states allows for the use of mathematical tools, such as linear algebra, to describe and analyze quantum systems. This approach also allows for the application of principles from classical mechanics, making it easier to understand and study quantum phenomena.
Unlike classical states, which can only have definite values for properties such as position and momentum, quantum states can exist in a superposition of multiple states. This means that a quantum system can have multiple possible outcomes when measured, whereas a classical system would have a single definite outcome.
The evolution of quantum states over time is described by the Schrödinger equation, which determines how the state vector changes in response to interactions with other systems. This evolution is continuous and deterministic, but the measurement process can cause the state to "collapse" into a definite outcome, in a way that is not fully understood.