Linked Questions
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What is the common difference between partial time derivative and ordinary time derivative? [duplicate]
What is difference between partial and ordinary time derivative?
for example: what is difference between $\frac {\partial v}{\partial t}$ and $\frac {dv}{dt}$?
where the $v$ is velocity.
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Why there is added a partial time derivative in formula for time derivative of potential energy? [duplicate]
In proving the total energy in conservative field is constant we have this equation(picture) why it added partial derivative? Why? I mean where it did come from?
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Difference between $dM/dt $ and $\partial M/\partial t$ [duplicate]
$\frac{dM}{dt} = 0$ represents a constant of motion $M.$ Why not $\frac{\partial M}{\partial t}$ represent a constant of motion $M$?
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How to explain implicit time dependence to someone? [duplicate]
I am trying to explain what implicit time dependence is and how it differs from explicit time dependence, but I'm unsure how "sound" my explanation is. Here is what I said:
Suppose I have a function $...
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Symbols of derivatives
What is the exact use of the symbols $\partial$, $\delta$ and $\mathrm{d}$ in derivatives in physics? How are they different and when are they used? It would be nice to get that settled once and for ...
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Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
For an observable $A$ and a Hamiltonian $H$, Wikipedia gives the time evolution equation for $A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}$ in the Heisenberg picture as
$$\frac{d}{dt} A(t) = \frac{i}{\hbar} ...
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Hamilton equations from Poisson bracket's formulation
Referring to Wikipedia we have that the equation of motion for a $f(q, p, t)$ comes from the formula
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t} f(p, q, t) = \frac{\partial f}{\partial q} \frac{\...
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Conserved quantities and total derivatives?
I am having a bit of a crisis in understanding of the physical meanings of total derivatives.
When a quantity $\rho$ (be it a vector or a scalar) is said to be conserved, then (mathematically) $$\...
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Which Schrödinger equation is correct? [duplicate]
In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrödinger equation is
$$H\Psi(x,t) = i\hbar\frac{\partial}{\partial ...
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Conserved quantities commute with Hamiltonian -- don't understand proof
http://en.wikipedia.org/wiki/Constant_of_motion#In_quantum_mechanics
I understand the derivation to get
$$\frac{d}{dt}⟨\psi|Q|\psi⟩ = -\frac{1}{i\hbar}⟨\psi|[H,Q]\psi⟩ + ⟨\psi|\frac{d}{dt}Q|\psi⟩$$
...
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Meaning of time derivative of an operator
Today when my professor was deriving this equation:
$$\frac{\mathrm d\langle A\rangle}{\mathrm dt}=\frac{i}{\hbar}\langle\left[H,\,A\right]\rangle+\left\langle\frac{\partial A}{\partial t}\right\...
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Can the kinetic energy be a function of the position vector?
,I got one confusion when reading Goldstein's Classical Mechanics (page 20, third edition). After getting the equation
$$
\sum \left\{\left[\frac{\mathrm{d}}{\mathrm{d}t}{\left(\frac{\partial T}{\...
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Full time-derivative, Poisson brackets and Hamilton's equations (classical mechanics)
While studying Poisson brackets in classical mechanics and the derivation of $\dot{q_j}=\{q_j,H\}$ and $\dot{p_j}=\{p_j,H\}$ form of Hamilton's equations I encountered a surpsing identity, which led ...
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Why the Lagrangian doesn't have an explicit time dependence?
I have a simple question regarding an example presented by Leonard Susskind and George Hrabovsky in their book on Classical Mechanics The Theoretical Minimum. In page 151, they state:
"If there is ...
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Does time-independence of potential energy imply time-independence of the Hamiltonian in Quantum Mechanics?
Consider a quantum mechanical system for a particle with Hamiltonian $\hat{H}=\frac{\hat{p}^2}{2m}+\hat{V}$ where $\hat{V}$ is the potential energy operator. and now let us assume that $\hat{V}$ is ...
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