Undergrad Angular Momentum Hydrogen Atom Problem: Physically Explained When L=0

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In quantum mechanics, the angular momentum for a hydrogen atom is described by the equation L = √(l(l+1))ħ. When L = 0, it indicates that l must also be 0, resulting in a wave function that is spherically symmetric and lacks angular dependence. This spherically symmetric condition is not exclusive to hydrogen; it also applies to other potential energy systems, such as the 3D harmonic oscillator and spherical potential wells. In these cases, energy eigenfunctions with L = 0 maintain the same spherical symmetry. Understanding these properties is crucial for analyzing quantum systems with similar characteristics.
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In quantum mechanics hydrogen atom problem ##L=\sqrt{l(l+1)}\hbar##. What that means physically when ##L=0##?
 
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L = 0 is only possible when l = 0 which means that the wave function has no angular dependence and spherically symmetric.
 
And also for a particle moving in some potential energy other than the hydrogenic ##V\propto 1/r##, the energy eigenfunctions with ##L=0## are spherically symmetric. These systems include the 3D harmonic oscillator ##V(x,y,z) = C(x^2 + y^2 + z^2 )## and the spherical potential well where ##V(r) = 0## if ##r<R## and ##V(r) = V_0## when ##r\geq R##.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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