How does a quantum mechanical state 'look' to different observer

[somitra]

Hi all

I am a little disturbed by the above thought about transformation of quantum state(or it's discription) between different frames of references. The doubt can be translated for a two state system as follows:

Let an observer has decided about her(his) x, y & z directions. He now writes down the state of a spin half particle as:
c₊f₊ + c_-f_-

f₊ & f_- are the spin up and spin down states for the z direction.

suppose now he decides to turn his axes about the z axis clockwise by angle p. At this point everyone will tell you that the state vector now transforms to:
c₊e^ipf₊ + c_-e^-ipf_-

Now my doubt is this. A physical system is represented by a vector in a hilbert space. Rotation of coordinate axes does not change the state of the system, so it should be represented by the same vector even after the rotation. Why should the representation of the same state differ in two different frames. Why should the state vector transform. It's expectation values should.

What I have been thinking is this: First of all we do not observe the state vector at all. It's a mathematical construct. When representing a system by a hilbert space, any other hilbert space isomorphic to it is as good. The observer(??) is free to pick any of these. The two vectors given above are connected by an automorphism(of rotation about z axis) on the system's hilbert space and both vectors may be made to represent the same state, i.e either observer can choose either of the two vectors to describe his state. By doing this the form of Sx, Sy , Sz is altered. Now vector spaces are abstract object. For them to be of any use we must represent them by using matrices. Is this representation by matrices of state vectors unique, after we have fixed the base states.

Am I right in asserting that a state vector is defined uniquely only upto an automorphism of the hilbert space(provided this automorphism is applied to all the state vectors), not just a phase difference. Does this automorphism alter the matrix representation(of vectors and operators). Also do any of these ideas follow from representation theory(I know very little of it).

 

[eljose79]

Thank you for bringing up this interesting question about the transformation of quantum states between different frames of reference. I can understand your concern about the consistency and uniqueness of the state vector representation in different coordinate systems. Let me try to address your doubts and provide some insights on this topic.

Firstly, it is important to understand that the state vector in quantum mechanics is not a physical quantity that can be observed directly, but rather a mathematical construct used to describe the quantum state of a system. This means that it is not tied to any specific coordinate system or observer, but rather a general representation of the system's quantum state.

When we talk about transforming the state vector between different frames of reference, we are essentially changing the basis in which the vector is represented. This is similar to changing the coordinate system in classical mechanics, where the same physical object can be described using different sets of coordinates. In quantum mechanics, this change of basis is achieved through unitary transformations, which preserve the inner product and thus the physical properties of the system.

Now, coming to your example of a spin half particle, the state vector you have described is indeed the same in both the original and rotated coordinate systems. However, the representation of this state in terms of spin operators (Sx, Sy, Sz) will differ in the two coordinate systems due to the change of basis. This does not mean that the state itself has changed, but rather the way we describe and measure it has changed.

Furthermore, the representation of the state vector using matrices is not unique, as you have correctly pointed out. This is because we can choose different bases and operators to represent the same state, and these choices are not restricted to just phase differences. However, these different representations are all connected by unitary transformations and thus describe the same physical state.

In summary, the transformation of quantum states between different frames of reference is a natural consequence of the mathematical framework of quantum mechanics. It does not change the physical state of the system, but rather the way we describe and measure it. I hope this helps to clarify your doubts and provide a deeper understanding of this topic.