Understanding Vector Components in Quantum Mechanics

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Homework Help Overview

The discussion revolves around understanding vector components in the context of quantum mechanics, specifically relating to a problem involving the addition of vector components. The original poster is attempting to clarify the relationship between a vector and its components, particularly in relation to angles and directions.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the validity of the equation Ax Sin(theta) + Ay Cos(theta) = A, particularly whether A is a vector or a scalar. There is exploration of the implications of this equation in terms of vector addition and direction.

Discussion Status

There is an active exploration of the definitions and relationships between the vector A and its components, with participants providing insights and questioning the assumptions made in the original statement. Some participants suggest alternative formulations and clarify the roles of the unit vectors in the context of the problem.

Contextual Notes

Participants reference a specific problem from Griffiths' quantum mechanics textbook, indicating that the context may involve specific coordinate systems and vector representations relevant to quantum mechanics.

phrygian
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Homework Statement



I read how to solve a problem I am working on, and part of it deals with adding vector components. A is the vector, Ax is the x component, Ay the y component, and theta is the angle A makes from the y axis.

Homework Equations


The Attempt at a Solution



The solution involves using Ax Sin(theta) + Ay Cos(theta) = A.

I know it seems easy but I can't seem to figure out why this would be true?
 
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phrygian said:
The solution involves using Ax Sin(theta) + Ay Cos(theta) = A.

I know it seems easy but I can't seem to figure out why this would be true?

This actually doesn't make sense if A is a scalar.

When you add vectors you get vectors not scalars.

If you were referring to \left\|A_x Sin(\theta) + A_y Cos(\theta) \right\| =A then that would make more sense.
 
What I mean was Ax Sin(theta) + Ay Cos(theta) = A where A, Ax and Ay are vectors, I don't see where this relation comes from?
 
In what direction is A,if it is a vector ? The only way this would make sense is if A has the direction of a_rho in cylindrical coordinates.

So I have to ask... in what direction is the unit vector of A?
 
Actually, your original question doesn't make sense if A is a vector- for exactly the opposite reason! If Ax and Ay are the x and y components of vector A, then "Ax cos(theta)+ Ay sin(theta)" is a scalar and cannot be equal to the vector A.

Assuming that this is in two dimensions, and vector A makes angle theta with the x-axis, then what is true is that Ax= |A|cos(theta) and Ay= |A| sin(theta) where |A| is the length of the vector A. You could also write that as "|A|cos(theta) i+ |A|sin(theta) j= A" where i and j are the unit vectors in the directions of the x and y axes respectively.
 
It's problem 4.50 in griffiths quantum mechanics and here is a quote from the solution manual:

We may as well choose axes so that a lies along the z axis and b is in the xz plane. Then S(1)a= S(1)z , and S(2)b = cosθ S(2)z + sinθ S(2)x .

S(1)a means the spin operator of particle 1 in the direction a.
 

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