# Tagged Questions

Numbers of the form $\{z= x+ i\,y:\;x,\, y\in\mathbb{R}\}$ where $i^2 = -1$. Useful especially as quantum mechanics, where system states take complex vector values.

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### About the complex nature of the wave function?

1. Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an ...
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### QM without complex numbers

I am trying to understand how complex numbers made their way into QM. Can we have a theory of the same physics without complex numbers? If so, is the theory using complex numbers easier?
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### Can one do the maths of physics without using $\sqrt{-1}$?

The use of imaginary and complex values comes up in many physics and engineering derivations. I have a question about that: Is the use of complex numbers simply to make the process of derivation ...
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### Why treat complex scalar field and its complex conjugate as two different fields?

I am new to QFT, so I may have some of the terminology incorrect. Many QFT books provide an example of deriving equations of motion for various free theories. One example is for a complex scalar ...
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### What does the complex electric field show?

We have a complex electric field. Is there any definition for absolute and imaginary part of a complex electric field? What do they stand for?
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### Born's Rule, What is the Reason? [duplicate]

As far as I've read online, there isn't a good explanation for the Born Rule. Is this the case? Why does taking the square of the wave function give you the Probability? Naturally it removes negatives ...
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### The meaning of imaginary time

What is imaginary (or complex) time? I was reading about Hawking's wave function of the universe and this topic came up. If imaginary mass and similar imaginary quantities do not make sense in physics,...
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### Special relativity and imaginary coefficient of the time coordinate

I read somewhere that part of Minkowski's inspiration for his formulation of Minkowski space was Poincare's observation that time could be understood as a fourth spatial dimension with an imaginary ...
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### Finding Stagnation Points from the complex potential

I am trying to find the stagnation point of a fluid flow from a complex potential. The complex potential is given by $$\Omega(z) = Uz + \cfrac{m}{2\pi}\ln z.$$ From this I found the streamfunction to ...
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### Is there a way to prove that a bound state wavefunction can always be chosen real for an arbitrary potential in Quantum Mechanics?

As we can prove many things that always (at least in introductory quantum mechanical problems) apply using an arbitrary potential (like that $E>V_{\rm min}$ or else the solutions are non-...
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### Schrödinger's Equation and its complex conjugate

I would like to know why there is a minus sign on the right-hand side of the SchrÃ¶dinger's complex conjugate equation, whereas in the SchrÃ¶dinger's equation there isn't. I know it is a simple question,...
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### Principal value of 1/x and few questions about complex analysis in Peskin's QFT textbook

When I learn QFT, I am bothered by many problems in complex analysis. 1) $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ I can't understand why $1/x$ can have a principal value ...
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### Applications of analytic continuation to physics

I posted this on math.SE, but didn't get much response. It might fit better on this site. Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the ...
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### Is there a direct physical interpretation for the complex wavefunction?

The Schrodinger equation in non-relativistic quantum mechanics yields the time-evolution of the so-called wavefunction corresponding to the system concerned under the action of the associated ...
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### Is quantum tunneling related to imaginary time?

I was studying for my exam and looking at the chapter which talks about Potential-energy graphs. Let's take this as an example: My book states that: "If the object is in $B$ and has a total energy ...
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### Motivating Complexification of Lie Algebras?

What is the motivation for complexifying a Lie algebra? In quantum mechanical angular momentum the commutation relations $$[J_x,J_y]=iJ_z, \quad [J_y,J_z] = iJ_x,\quad [J_z,J_x] = iJ_y$$ become, on ...
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### What does the Schrodinger Equation really mean?

I understand that the Schrodinger equation is actually a principle that cannot be proven. But can someone give a plausible foundation for it and give it some physical meaning/interpretation. I guess I'...
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### Stokes' theorem in complex coordinates (CFT)

I am studying CFT, where I encounter Stokes' theorem in complex coordinates: $$\int_R (\partial_zv^z + \partial_{\bar{z}}v^{\bar{z}})dzd\bar{z} = i \int_{\partial R}(v^{z}d\bar{z} - v^{\bar{z}}dz).$$...
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### Why complex functions for explaining wave particle duality?

I have this very bad habit of going to the scratch, discarding all the developments of a theory and worldly knowledge, and ask some fundamental (mostly stupid and naive, as some may say) questions as ...
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### Quaternions and 4-vectors

I recently realised that quaternions could be used to write intervals or norms of vectors in special relativity: $$(t,ix,jy,kz)^2 = t^2 + (ix)^2 + (jy)^2 + (kz)^2 = t^2 - x^2 - y^2 - z^2$$ Is it ...
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### QFT Hilbert spaces over other rings than the complex numbers $\mathbb{C}$

I would like some help evaluating a physics theory recently proposed by a physics professor at the College of Dupage. I think the theory is utterly wrong, for very simple reasons. If an amateur ...
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### Is the step of analytic continuation unavoidable or can you model around it?

One sometimes considers the analytic continuation of certain quantities in physics and take them seriously. More so than the direct or actual values, actually. For example if you use the procedure ...
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### Confused over complex representation of the wave

My quantum mechanics textbook says that the following is a representation of a wave traveling in the +$x$ direction:$$\Psi(x,t)=Ae^{i\left(kx-\omega t\right)}\tag1$$ I'm having trouble visualizing ...
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### Born Interpretation of Wave Function

I have just started Griffiths Intro to QM. I was studying Born's interpretation of Wave function and it says that the square of the modulus of the wave function is a measure of the probability of ...
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### Concrete example of the application of complex analysis in electrostatics [closed]

I've heard complex analysis can be useful in solving electrostatics problems, but despite doing some research I was unable to find any concrete examples. Would anyone be able to provide a simple ...
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### Complex numbers in optics

I have recently studied optics. But I feel having missed something important: how can amplitudes of light waves be complex numbers?
There are 2 parts to my question: 1) Say we choose the metric signature to be (-+++), as in the Wikipedia page. Then the invariant interval in Minkowski space is written: $ds^{2} = -(dt^{2}) + dx^{... 0answers 328 views ### Probability and probability amplitude [duplicate] What made scientists believe that we should calculate probability$P$as the$P = \left|\psi\right|^2$in quantum mechanics? Was it the double slit experiment? How? Is there anywhere in the ... 2answers 390 views ### Imaginary number for extinction coefficient in complex refractive index In complex refractive index on a material,$n=n'+ ik$, the imaginary part$k$is physical meaning, as it shows absorption in the material but it is an imaginary. How we measure an imaginary values in ... 5answers 4k views ### Complex Conjugate of Wave Function I've been reading through Griffiths QM book, and the only thing bugging me is they never fully described what$\Psi^* $should be for any given function. I know it's the complex conjugate at the same ... 1answer 314 views ### What is the advantage of using exponential function over trigonometric function in analyzing waves? A.P.French in his book Vibrations and Waves writes: . . . Why should the exponential function be such an important contribution to the analysis of vibrations? The prime reason is the special ... 2answers 155 views ### Quantum Mechanical States What can be the precise answer to the question that Quantum states are complex and infinite dimensional. Why is this so? Is it because they belong to the complex Hilbert space? Even if they ... 6answers 3k views ### What does imaginary number maps to physically? I am taking undergraduate quantum mechanics currently, and the concept of an imaginary number had always troubled me. I always feel that complex numbers are more of a mathematical convenience, but ... 2answers 417 views ### Why do we must initially assume that the wavefunction is complex? The sound waves are real, and they can interfere, so corresponding apparat may be used in quantum mechanics. We also may use the time dependence in a form of orthogonal matrix multiplying the initial ... 4answers 2k views ### Relativistic mass and imaginary mass The (relativistic) mass of an object measured by an observer in the$xyz$-frame is given by $$m = \frac{m_{rest}}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}.$$ Mathematically$v\$ could be greater than the ...
My question doesn't go much beyond the title: Why does $$\left | \psi \left ( x,t \right ) \right |^{2}$$ give us the probability density of something appearing at a certain location? I understand ...