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### How can we ignore the diverging term $e^\infty$ in the integral?

You can rewrite the indefinite integral as $$e^{-a x} f(x)$$ where $f(x)$ does not grow exponentially with $x$. (In fact $f(x) \sim e^{\pm i k x}$ is an oscillating ...
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### Deriving Schrödinger equation from Ehrenfest Theorem

This step is a mathematical one; in general, if $$\int f = \int g,$$ you can't conclude that $f=g$. But if $$\int f \psi = \int g \psi$$ for a large class of functions $\psi,$ then you sometimes ...
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### Quantum mechanics of a moving bound state in the lab frame

Actually, your two examples here are subtly different: Consider a non-relativistic quantum-mechanical problem of two bodies interacting via a confining potential 𝑉(𝐫⃗ ) (say, a hydrogen atom) ...
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### Are the shapes of atomic orbitals direct consequence of the Schrödinger equation?

I am trying to understand whether the shapes of the orbitals are inevitable given the standard model. The atomic electron orbitals of atomic physics come about because the atomic potential is ...
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### Time dependent Schrodinger equation through variation principle - questions about derivation

Well, let's see. The Lagrangian is $$L~=~ \frac{\langle \psi | i \partial_t -H |\psi\rangle}{2||\psi ||^2} ~+~ c.c.,\qquad ||\psi ||^2~\equiv~\langle \psi | \psi\rangle. \tag{A}$$ An infinitesimal ...
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### Definition of transmission and reflection coefficients for a particle

In general, $R$ and $T$ can be defined with probability current. In 1D, $$j = \frac{\hbar}{ 2mi} ( \psi^* \frac{\partial \psi}{\partial x} - \psi \frac{\partial \psi^*}{\partial x} )$$ In 3D, partial ...
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### Radial Schrödinger equation: from $R_l(r)$ to $u_l(r)$

While it seems like you've resolved your particular issue, this might be a good opportunity to think about how operations work more generally, as you will come to see a lot of operator algebra in ...
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### Feynman path integral in an EM field

In principle the Feynman fudge factor (4-8) (which comes from Gaussian momentum integrations of the Hamiltonian phase space path integral, cf. e.g. this Phys.SE post) should not be affected by the E&...
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