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Questions tagged [quantum-mechanics]

For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales.

Quantum mechanics, also known as quantum physics or quantum theory, deals with classical physical phenomena at quantum scales.

The precise nature of the subject has changed over the years. This article explains its current formulation.

Quantum mechanics aims toprovide a mathematical description of the dual particle-like and wave-like behaviors and interactions of energy and matter.

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Applying Variational Approximation

I am trying to solve a problem involving variational approximation, where the task is to calculate a value $C$ such that $C > \frac{\int_{-\infty}^{\infty} |f'(y)|^2 dy}{\int_{-\infty}^{\infty} \frac{1}{y^2} |f(y)|^2 dy}$ for a given function $f$…
Newbie
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Expectation value of $x^2$ in the quantum harmonic oscillator

I'm trying to show that $$\dfrac{1}{2^n \ n! \ \sqrt{\pi}}\int_{-\infty}^{\infty}x^2 e^{-x^2}H_n(x)^2 = n+\dfrac{1}{2}.$$ It's the expectation value of $x^2$ in the quantum harmonic oscillator. Does anyone have a hint here? Some propertie of the…
fourier
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What does $|\Psi\rangle_S, |\Psi\rangle_H$ mean in bra-ket notation?

Could anyone familiar with bra-ket notation help explain the generalized Schrodinger equation: $$ H |\Psi(t)\rangle_S = i \frac{d}{dt}|\Psi(t)\rangle_S$$ One can therefore write: $$|\Psi(t)\rangle_S = e^{-iHt}|\Psi\rangle_H$$ What do the…
James
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Why do we need equal number of constraints and parameters to have a solution?

While going through Shankar's Principles of Quantum Mechanics (2nd edition), in the Chapter 5 at page 179, while explaining the concept of quantization of energy for particle in a box he argued (though I may be wrong) that we need equal amount of…
Soumyadeep Chakraborty
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How does CNOT behave for superposition control states?

Descriptions of the quantum conditional not (CNOT) gate can be very terse. I would like some help so that I can understand this gate more clearly. When the control qubit is |0> or |1>, then the behavior of CNOT on the target qubit is easy to…
mcandre
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What Condition is Condition for $\Omega$ that it will define a inner-product?

What is the condition on the $N\times N$ Matrix $\Omega$, So that $Tr\left[{A}^{\dagger}B\Omega\right]$ is a valid inner product in Vector Space of $N\times N$ Complex Matrices?(Where $A$ and $B$ are $N\times N$ Matrix) I tried by Matrix…
Suman Mandal
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Normalization and square integrables in quantum mechanics

The sum of two square integrable function is itself square integrable but sum of two normalized functions is not generally normalized. Why?
Mahavir Mahto
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Resolution of identity for the multiplication operator

I'm starting my studies in spectral theory and my professor tells me that I should study the resolution of identity. I thought I understood the theory, but when I arrive at the examples ... I can't understand them. So I really appreciate if someone…
May Souza
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Form of scattering solutions for a square potential barrier

Consider an incoming beam of particles with $E>U$ scattering from the potential $V(x) = \begin{cases} 0 & \text{ for } x<-a\\ U & \text{ for } |x|a \end{cases}$ So Schrodinger's (time-independent) equation reads $\psi'' =…
Vadim
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How to calculate the overlap of four cartesian gaussian function i.e $\int \phi_{A}(r)\phi_{B}(r)\phi_{C}(r)\phi_{D}(r)d^{3}r$?

I want to calculate $\int \phi_{A}(r)\phi_{B}(r)\phi_{C}(r)\phi_{D}(r)d^{3}r$ where $\phi_{A}(r)=(x-A_{x})^{l1}(y-A_{y})^{l2}(z-A_{z})^{l3}e^{-\alpha |r-A|^{2}}$. l1, l2 and l3 are integers Can you suggest any book and any code in c++/matlab that…
Sayan Das
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Uncertainty Principle For a Generic State

Firstly, I know this is a physics problem, but this forum is so much more active, and I'm sure someone here could help me. The problem is the following: Given a spin-1/2 particle, and the generic state: $$ |\psi\rangle =…
H44S
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Solving differential equation

I have the following differential equation \begin{align} \left[-\partial^{2}_{x}+ x^{2} -1-2\alpha_{+}\right] \left[-\partial^{2}_{x}+ x^{2} -1-2\alpha_{-}\right] \phi(x)=0, \quad \alpha_{\pm}=\dfrac{-1}{2}+\dfrac{E^{2}}{2}\pm…
Elmanara
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Power series of Hermite polynomial generating function

The generating function for Hermite polynomials is: $e^{-s^2+2s\xi} = \sum_{n=0}^{\infty}H_n(\xi) \frac{s^n}{n!}$. How does one do the power series expansion for $e^{-s^2+2s\xi}$?
David Kwak
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How to get $E$ (Eigenvalue) or $E_n$ from this transcendental complex equation?

I want to solve this transcendental complex equation, but I don't know the step by step to get E in form equation and value of E f[E] = [exp^(i*p*b)/(16*A^2*B*H) {((A + H)^2)[(A + B)^2*exp^(2*i*(k (a - b) - q*a)) - (A - B)^2*exp^(-2*i*(k (a - b) -…
Aff_
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Commutation relations in quantum mechanics?

I'm looking for proof of the following commutation relations, $ [\hat{n}, \hat{a}^k] = -k a^k, \quad \quad \quad \quad [\hat{n}, \hat{a}^{\dagger k}] = -k \hat{a}^{\dagger k} $ where $\hat{n}$ is the number operator $\hat{n} = \hat{a}^{\dagger}…
Advaita
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