Tagged Questions

In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

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Question about Eigenvalues of Hermetian Operators Being Real Numbers

I'm still slogging through Quantum Mechanics: The Theoretical Minimum and I've reached another area that baffles me. Susskind uses the following to show that the eigenvalues of Hermitian operators ...
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Entangled wave function and polarisation operator

I was working on the following problem from Quantum Chemistry and Spectroscopy by T. Engel (3rd Edition), and was stumped in a few places. I wish get some feedback on my solution The problem is the ...
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Hello I am currently studying introductory QM and am confused about bases and operators. If I have an operator $\hat{Q}$, does this represent a change of basis matrix? In other words, does $\hat{Q} | \... 2answers 6k views What is the Momentum Operator? I know the equation for the momentum operator, but what exactly is the momentum operator? It's bizarre to me that taking the derivative of the wave function, which is an operator, should return ... 1answer 68 views Number operator in quantum mechanics In quantum mechanics$a^{\dagger}a$is defined as the number operator, where$[a,a^{\dagger}]=1$. Why cannot we define$aa^{\dagger}$as number operator instead of the usual definition? 2answers 2k views Eigenstates of a shifted harmonic oscillator Let's say I have a quantum harmonic oscillator$H = \omega a^\dagger a$, where$a^\dagger$is the raising operator and$a$is the lowering operator and$H |n\rangle = \omega n |n\rangle$. Now assume ... 1answer 47 views Verification of proof of complete set of commuting operators Hi I am interested in the validity of the following proof. I am interested in the validity of this particular proof as I am aware of how to prove this result in a different way. Theorem: If two ... 1answer 60 views Expectation value in second quantization I am stuck calculating a simple expectation value for an operator, which is expressed in second quantization. I know the result, but I fail to proof it. Lets say I have one-particle wave function$|\...
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So in my last question, @joshphysics showed me how to prove $K_\pm$ were ladder operators. Now I need to show that there is a lowest state, i.e $$\langle m_0|K_+=K_-|m_0\rangle=0$$ I am not ...
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Calculating eigenvalues for operator [closed]

Given relation $[a,a^\dagger]=I$. Operator $K$ is defined as $K=a^\dagger a+\lambda a^\dagger+\lambda^* a$. I need to find the eignevalues of operator $K$. How realtion that involves commutator could ...
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Non-separable solutions of the Schroedinger equation

I'm studying an undergraduate Quantum Mechanics course and I have some doubts about the solution of the Schroedinger equation by the separation of variables method. If we suppose that the solutions ...
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Why is Wick contraction a $c$-number?

It is mentioned in Fetter's Quantum Theory of Many-Particle Systems (in contraction part of section 8 Wick's Theorem), that: contractions are c numbers in the occupation-number Hilbert space, not ...
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Glauber formula, Baker

somewhere I read here: $[A,F(B)]=[A,B]F'(B)$ is used to prove Glauber's formula $\exp(A+B).\exp([A,B]/2)$ I have tried and looked everywhere to try and understand this to no avail. The first is in ...
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Simple QFT simulation - how to do it

I would like to write a simple QFT simulation for a free scalar field with a cubic interaction term. However, I got stuck a bit. I will try to describe what I think I understand. I want to have a ...
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Where does $\hat{P}\psi(x) = -i\hbar \partial_x \psi(x)$ come from?

It's a very basic question, where does the relation $$\hat{P}\psi(x) = -i\hbar \partial_x \psi(x)$$ for any square integrable $\psi(x)$ come into existence? Some texts I found states that the above ...
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Rigorous definition of density of states for continuous spectrum

For operators with pure point spectra it is clear how to count number of states corresponding to a given eigenvalue - one can just calculate the dimension of eigenspaces. I am wondering how to do it ...
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The Momentum Operator in QM

I've seen the 'derivation' as to why momentum is an operator, but I still don't buy it. Momentum has always been just a product $m{\bf v}$. Why should it now be an operator. Why can't we just multiply ...
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OPE coefficients identity operator

when I have two canonically normalized operators $\phi_{1}$ and $\phi_{2}$ and I want to compute their OPE in terms of the identity operator, is there any way to actually calculate the first levels of ...
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Parity operators and spin

Consider the following excerpt from Weinberg's Lectures on Quantum Mechanics: I follow everything up until the last statement in the excerpt. In fact, from other things I've read, it seems that one ...
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Does Heisenberg's uncertainty hold for any two quantum measurements?

Heisenberg's uncertainty principle is most commonly expressed in terms of the uncertainty in measurement of position and momentum of a particle, $$\Delta x\Delta p \geq \hbar$$and uncertainty in ...
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Taylor expansion of exponential operator

I have an operator: $$\hat O = e^{\hat A+\hat B}$$ Is it correct to write its first order Taylor expansion by: $$\hat O = 1+\hat A+\hat B$$
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Uncertainty Principle with the corresponding operators

Why does the corresponding operator do not commute if there is uncertainty related to two observables A and B that states $\Delta A\,\Delta B > 0$ ?