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Results tagged with quantum-mechanics
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user 21270
Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.
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Hamiltonian Expectation Value for a Non-Local Potential
For a Hamiltonian $\hat{H}$ where the potential energy operator $\hat{V}$ is local, the expectation value of $\hat{H}$ with respect to a momentum space state ket $|\phi_p\rangle$ is given by
$$ \lang …
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1
answer
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Sinusoidal drive of two level system: why can we ignore one of the two terms?
I am studying the time-dependent perturbation theory from Griffith's Introduction to Quantum Mechanics. The context is a two-level system under a sinusoidal perturbation: $H'(\textbf{r}, t) = V(\textb …
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Total Angular Momentum of a Hydrogen Atom
Griffiths in his celebrated book named 'Introduction to Quantum Mechanics' discusses about the total angular momentum of a hydrogen atom on page 187.
He writes:
If a hydrogen atom is in the stat …
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1
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Introducing a New Quantum Number $l$ for Bloch Functions
If we solve the time-independent Schrödinger equation in 1D
$$\frac{d^2\psi(x)}{dx^2} + \frac{2m}{\hbar^2} \left[E - V(x)\right] \psi(x) = 0$$
for a periodic potential $V(x)$ with wave function $\ps …
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Expression for Delta Function in Quantum Mechanics
For a free particle, we can derive the following well-known relation:
$$\langle k|k'\rangle = \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-i(k-k')x} \, dx = \delta(k-k'). \tag{1.10.33}$$
Reference: …
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State Vector in the Schrodinger Equation
Let us consider a nonrelativistic particle of mass $m$, spin $s$ and isospin $i$. The Schrodinger equation for the state vector $|\psi(t)\rangle$ of this particle is given by
$$i\hbar\frac{d|\psi(t)\ …
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How do I evalute $\langle n|x^2 |n\rangle$ using the annihilation and creation operators?
I am assuming that the question is in the context of $1D$ simple harmonic oscillator.
If you consult any introductory quantum mechanics textbook, you will see that $\hat{x}$ can be written as $K(\ha …
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Scattering by a Delta Function Well in 1D [closed]
Let us consider a scattering process by a delta function well in 1D:
$$
V(x) = -\alpha \, \delta(x), \quad \alpha > 0.
$$
I solve the Schrödinger equation for the scattering states and get the follow …
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Parity and the Dipole Operator for a Two-Level System
Answer to the First Question
I will show that $\langle a \vert \textbf{r}_e \vert a \rangle = 0$, which also answers my first question.
\begin{eqnarray}
\langle a \vert \textbf{r}_e \vert a \rangle …
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Recoil of an atom absorbing a photon
I am reading the article named Manipulating Atoms with Photons by Claude Cohen-Tannoudji and Jean Dalibard. On page no. 16, the following is said.
Consider the atom in its ground state $g$ and its ce …
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1
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Parity and the Dipole Operator for a Two-Level System
I attempt to understand the parity and dipole operator from Daniel Steck's notes: Quantum and Atom Optics (page no. 152, section 5.1.1). I have also attached a screenshot at the end of the question.
…
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Eigenkets in Interaction Picture
Let us consider a system. In Schrodinger picture, its Hamiltonian $H$ is given by $H = H_0 + V(t)$, where $H_0$ is the unperturbed Hamiltonian and $V(t)$ is the time-dependent perturbation.
In intera …
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Eigenkets in Interaction Picture
The state kets and operators in the interaction picture are defined via a unitary transformation:
$$
|\psi(t)\rangle_I = e^{iH_0t} |\psi(t)\rangle_S, \quad A_I(t) = e^{iH_0t} A_S \, e^{-iH_0t}
$$
wh …
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$k$-interval for First Brillouin Zone
If we solve the time-independent Schrodinger equation in 1D for a periodic potential $V(x)$ with wave function $\psi(x)$ subject to a periodic boundary condition: $\psi(x) = \psi(x + Ga)$, where $a$ …
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Particle in a Box: Energy Less than the Potential Energy
I am reading quantum mechanics from Shankar's Principles of Quantum Mechanics. On page 157 he defines the box potential $V(x)$ as
$$
V(x) = \left\{ \begin{array}{rl}
0 &\mbox{ if $|x|< L/2$} \\
\inft …
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