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The equation will become $$ - \frac{\hbar}{2m} \bigg( \frac{\partial^2}{\partial x^2} + \frac{ \partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \bigg) \psi(x,y,z) = (E-V_0) \psi(x,y,z) $$ And the solutions are the same: $$ \psi_{n_x, n_y, n_z} (x,y,z) = C \sin( \frac{n_x \pi x}{L}) \sin (\frac{n_y \pi y}{L}) \sin(\frac{n_z \pi z}{L}). $$ And ...


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The motion of electrons in the wires and the voltages can't be "seen" by naked eyes so the whole science of electric circuits is automatically "harder to visualize" than mechanics. But all such laws and phenomena have mathematically similar analogies in mechanics. The voltage is analogous – not only mathematically but physically – to the slope of an ...


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Start by noting that the electrical potential is an energy per unit charge. In an electric field $E$ the field produces a force on a charge $Q$ of: $$ F=EQ $$ so if we move the charge a distance $dr$ the work done is just force times distance or: $$ W=EQ\,dr $$ The work done per unit charge is $E\,dr$, and this is what we mean by the change in the ...


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Gravitational field is a vector field and is determined by negative gradient of the gravitational potential. $$\vec g=-\vec\nabla \phi$$ Frome equation above, it is obvious that $|\vec g|=|-\vec\nabla \phi|$ (magnitude of $\vec g$ is equal to magnitude of $-\vec \nabla \phi$) and we know that $|-\vec\nabla \phi|$ is a non-negative quantity. You have made a ...


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Short (but cryptic) answer: complex numbers arise in quantum mechanics because we would like find solutions to the differential equation $$\frac{\partial}{\partial x}f(x) = cf(x)$$ which don't blow up as $x\to \pm\infty$. Long answer: Fundamentally, the shift from classical mechanics to quantum mechanics is replacing functions (observables) and numbers (...


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The values of $E_c$ and $E_v$ in the band diagram depend on the point of reference. So yes they can have negative values if you chose your reference that way. Keep in mind that their difference $E_g$ stays constant nonetheless. Electrons are fermions and therefore governed by Fermi-Dirac statistics. That means that they have to comply with the Pauli ...


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Charges at rest move when a force is applied on them and this is due to Newton's laws. Now to apply a force, we need a field, like electric/gravitational field. Each field acts upon certain measurable properties of a system, like gravitational on mass, electric on charge etc. Now potential is just a fancy name of height in electromagnetism. I hope you're ...


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Let us say plate A has a charge q1 and plate B, which faces plate A has a charge q2. By making use of the fact that the net field in the bulk of a conductor in static conditions is zero, and that the net field near the outer surface of a conductor equals [local surface charge density/€0], you can prove the following: Charge on the outer surfaces of A and B ...



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