How Do Complex Numbers Affect Ket Vectors in Quantum Mechanics?

[Einstein's Cat]

In Dirac's "The Principles of Quantum Mechanics," ket vectors are multipled by complex numbers (c1 |A> + c2 |A> = c1 + c2 |A>) and I was curious what affect this has a) on the ket vector and b) on the entire system? Also is (c1 |A> + c2 |A> = c1 + c2 |A>) equal to (|A> + |A> = |A>)? Thank you for your help!

 

[Heinera]

At this level, just think about ket vectors as ordinary column vectors from any introductory text on linear algebra.

a) If you multiply a vector by a complex scalar, it means you multiply each component in the vector by the scalar.

b) No, |A> + |A> = 2|A>.

 

[Doc Al]

In Dirac's "The Principles of Quantum Mechanics," ket vectors are multipled by complex numbers (c1 |A> + c2 |A> = c1 + c2 |A>) and I was curious what affect this has a) on the ket vector and b) on the entire system?

As far as the ket itself goes, multiplying it by some complex number has no physical significance. c1 |A> and (c1 + c2) |A> would refer to the same state. (For mathematical convenience, we usually normalize the state.)

When you are composing a state out of several different kets, then those coefficients are significant as they reflect the relative phase of each ket in the overall state.

Also is (c1 |A> + c2 |A> = c1 + c2 |A>) equal to (|A> + |A> = |A>)?

I think you missed a factor of 2 there.

 

[Einstein's Cat]

Thank you very much for your time; it's very helpful.