Complex vectors Definition and 27 Threads

I'd like to discuss some aspects of quantum systems' state space from a mathematical perspective. Take for a instance a qubit, e.g. a two-state quantum system and consider the set of its pure states. This set as such is a "concrete" set, namely the "bag" containing all the qubit's pure states...

Hello! 1) I read about complex vectors and I tried to imagine how it would look like in coordinate system. I was confused because complex vector consist of two vectors ##\vec a## and ##\vec b## : ##z=\vec a+ \vec bi##. I know that this complex vector can be visualized in 4D space. But I can't...

This is problem A.4 from Quantum Mechanics – by Griffiths & Schroeter. I cannot make the Gram-Schmidt procedure work. I don't know whether I am just inept with complex vectors or I have made some wrong assumption. The Gram-Schmidt procedure (modified, I think) Suppose you start with a basis...

Summary:: summation of the components of a complex vector Hi, In my textbook I have ##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}## ##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}## For ##\hat{e_p} = \hat{x}##...

Hi Here is my attempt at a solution for problems 1) and 2) that can be found within the summary. Problem 1) a = 3-2i b= -6-4i c= 4+ 6i d= -4+3i Now, to calculate each vector modulus, I applied the following formula: $$\left| Vector modulus \right| = \sqrt {(a^2 + b^2) }$$ where a = real part...

Homework Statement Homework EquationsThe Attempt at a Solution since they are in complex vector form, its missing a " j " on the sin() part, correct?

Homework Statement let \epsilon_1 and \epsilon_2 be unit vectors in R3. Define two complex unit vectors as follows: \epsilon_{\pm} = \frac{1}{\sqrt{2}}(\epsilon_1 \pm i \epsilon_2) verify that epsilon plus minus constitutes a set of complex orthonormal unit vectors. That is, show that...

Homework Statement Let a is a complex vector given by a = 2π K - i ρ / α^2 , where ρ is a two dimensional position vector and K is the corresponding two dimensional vector in the Fourier space. In order to find magnitude of this vector, i found that it is 4π^2 K^2 + ρ^2 / α^4 . The logic...

So I was trying to learn how to find the angle between two complex 4-dimentional vectors. I came across this paper, http://arxiv.org/pdf/math/9904077.pdf which I found to be a little confusing and as a result not overly helpful. I was wondering if anyone could help at all? Many thanks in...

Hi everyone, I am interested how is polarized light absorbed by a molecule or an atom. Unfortunately, I come to a problem in the derivation where a complex vector in a real space appears. This is something I never seen before and I do not know how to interpret it. Therefore I would like to ask...

The way I understand it, they both have rectangular forms which are easy for addition/subtraction. Now I realize that the polar form of a complex vector can be simplified into an exponential, which is ideal for multiplication/division. But this is what confuses me; vectors don't multiply/divide...

I've recently began a course on electromagnetism and have started dealing with complex vectors. I have a couple questions to ask: Regarding the general concept of complex vectors, I am curious what these actually represent. Refer to attached equation. Am I correct to believe that this...

I was reading in my textbook that the scalar product of two complex vectors is also complex (I assuming this is true in general, but not in every case). However for the general definition (the inner product), each element of one of the vectors needs to be its complex conjugate. I learned this...

How is computed the cross product of complex vectors? Let ##\mathbf{a}## and ##\mathbf{b}## be two vectors, each having complex components. $$\mathbf{a} = a_x \mathbf{\hat{x}} + a_y \mathbf{\hat{y}} + a_z \mathbf{\hat{z}}$$ $$\mathbf{b} = b_x \mathbf{\hat{x}} + b_y \mathbf{\hat{y}} + b_z...

I have three (N x 1) complex vectors, a, b and c. I know the following conditions: (1) a and b are orthonormal (but length of c is unknown) (2) c lies in the same 2D plane as a and b (3) aHc = x (purely real, known) (4) bHc = iy (purely imaginary, unknown) where (.)H denotes...

I need some clarification on projections of complex vectors. If I have a nxm matrix of complex numbers V and a mx1 matrix s, and I want to find the projection of s onto any column of V. The formula to do this is c = <V, s>/||V(j)||^2 where V(j) is the column of V to be used. My question is...

Homework Statement I have to plot the following complex numbers into the complex plane: X_C=0-1350i X_L=0+980i R_L=100+0i

I was wondering, is the definition of the cross product the same for complex vectors? And if it is, then how is its geometric interpretation, that is ||\mathbf{a}\times\mathbf{b}||=||\mathbf{a}|| \; ||\mathbf{b}|| \sin\theta Thanks.

Would you pls help me with the following vector product? I got no idea how the author derived the second equation from the first one. My derivation result is always including the imaginary unit i for the second term in the second equation on the right hand side. Specifically, how to verify that...

Homework Statement Let z, w be complex vectors of C^n. Prove ||w + z|| <= ||w|| + ||z|| (using the standard inner product for C^n) (i.e. <w,z> = w*z', where * is the dot product and ' denotes the complex conjugate) The Attempt at a Solution Well, I found that ||w + z|| =...

What is the relationship btw the Hermitian inner product btw 2 complex vectors & angle btw them. x,y are 2 complex vectors. \theta angle btw them what is the relation btw x^{H}y and cos(\theta)?? Any help will be good?

Statement: <v(t)> = \frac{1}{T} \int^{T}_{0}v(t)dt = \frac{1}{T} \int^{T}_{0}V_0cos(\omega t + \phi)dt \equiv 0. (#1)Relevant Question: If we suppose v(t) is a complex vector, is the second equality above still true?Reasoning: If v(t) is a complex vector, then v(t) = cos(\omega t)\hat{x} +...

Homework Statement Consider the unit vector, \hat{v}(t), expressed in instantaneous form: \hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y} (#0) The Vector will rotate counterclockwise in the x-y plane with angular velocity \omega. Since both components are sinusoidally time varying...

Problem/Statement The complex vector, \hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y} is the unit vector \hat{v}(t) expressed in instantaneous form. Question What I am wondering is, why is there no imaginary component "j" in say the sin component for the equation above? Can we...

Homework Statement Can someone explain to me why complex vectors of the following form will rotate counterclockwise in the x-y plane [with a velocity of w], \hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y} (#1) And why the following equation, the unit vector rotates in the clockwise...

Hello, I have recently started to study some Geometric Algebra. I was wondering how should I interpret complex-vectors in \mathcal{C}^n in the framework of Geometric Algebra. I understand already that a complex-scalar should be interpreted as an entity of the kind: z = x + y (\textbf{e}_1...

Homework Statement Find a basis for V=\mathbb{C}^1, where the field is the real numbers. The Attempt at a Solution I'd say \vec{e}_1=(1,0), \vec{e}_2=(i,0) is a basis, because it seems to me that \vec{u}=a+bi \in V can be written as...