# 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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### Why does Griffiths define the complex inner product differently? [closed]

I have just now noticed that Griffiths (in his book Introduction to Quantum Mechanics) defines the complex inner product as $\big<z,w\big>=\sum_{i=1}^n\overline{z}_iw_i$. In all mathematics ...
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### Why Hamiltonian is Hermitian? [duplicate]

Everyone knows that this is needed to make eigenvalues real, but still why we enforcing such a structure at first place? An arbitrary operator can have as complex as real eigenvalues, we can simply ...
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### How to interpret the complex index of refraction?

The index of refraction which represents how much light gets refracted when entering a medium is defined as $$n = \frac{c}{v}$$ I have seen it stated in several places, such as here, that we can ...
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### Causality and response functions

Referring to David Tong's notes on Electromagnetism, page 29 (of the PDF, numbered 183), section 7.5.4; It is proved that the frequency domain response function (in this case describing the ...
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### Can temperature be a complex number?

Is it possible for a temperature to be a complex number? I want to say "no" but I can't be so sure. If it is possible I would like to know of an example. I found an interesting article which treats ...
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### What is the purpose of the imaginary portion of the wave function?

I recently watched this video. I'm trying to learn about the origin of the wave function and therefore understand its use in the Schrödinger Equation. However at the end of the video I understood up ...
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### Why does the magnitude squared of the wave function give us the probability density? [duplicate]

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 ...
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### Why are complex fields in the Lagrangian?

I know that a complex field has twice the number of degrees of freedom of a real field, and that fields (in QFT) aren't observables so we don't really care if they are real. But why the need for ...
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### Metal Refractive index

I'm working on Fresnel equation for calculation of reflection of a light (532 nm) on Iron. I've got a question: Is metals refractive index always a real number or it can be a complex number?
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### QM: Why is there a minus sign on the Energy operator when using complex conjugate?

I understand how they get the first equation. But I have no idea why there is a minus sign on the second equation: This is from a derivation for the probability density current found here: http://...
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### Why is imaginary time “outdated”? [closed]

I was looking at reviews for Sakurai's Quantum Mechanics textbook, and some mentioned it being outdated, specifically mentioning his use of imaginary time. Is this idea deliberately avoided in modern ...
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### Can we define the zero potential at an imaginary point?

Consider a force field defined as $$\vec{F}(x) = \left(\frac{A}{x^2}-B\right)\hat{i}\space$$ where $A, B$ are positive constants. We want to get the potential energy function for this field. We can ...
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### $SU(2)$ generators and creation annihilation operators

The algebraic method to find the irreducible representation of the $SU(2)$ group makes use of the operators: $$J_z\\J_+=\frac{1}{\sqrt{2}}(J_x+iJ_y)\\J_-=\frac{1}{\sqrt{2}}(J_x-iJ_y)$$ In the book ...
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### Complex dimensional analysis

Does complex numbers have physical dimensions? Is it sensible to talk about the dimensional analysis of $Z$ where $Z$ is the impedance of a mechanical oscillatory system? Or is it the $|Z|$ which has ...
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### Hermitian Metric and Geodesics

Why isn't general relativity developed with a Hermitian metric and a theory of complex valued paths and geodesics? The concept of arc length and geodesic suffers under a pseudo-Riemannian metric. My ...
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### A question over the reality of $\sin x$

Harmonic functions are in widespread use in physical descriptions of natural real phenomena. I am just wondering therefore how we can define $\sin(x)$ to be part of a real physical quantity (with ...
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### Root of $i$, which one to take?

The propagator of a free particle in 1d is $$K(x_b, t_b; x_a, t_a ) = \sqrt{\frac{m}{2\pi i \hbar (t_b-t_a)}} \exp \left [ \frac{i m (x_b-x_a)^2}{2 \hbar (t_b-t_a)} \quad \right ] .$$ It looks nice....
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### Why is the value of the action integral in general relativity the same on all regions that are homologous?

In their famous paper Action integrals and partition functions in quantum gravity, Gibbons and Hawking argue that in order to avoid the singularity of a Schwarzschild black hole you can complexify ...
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 ...