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LagrangeEuler
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If Hamiltonian commutes with a parity operator ##Px=-x## are then all eigenstates even or odd? Is it true always or only in one-dimensional case?
Not necessarily. If the eigenstates are non-degenerate, then they must be even (edit or odd).LagrangeEuler said:If Hamiltonian commutes with a parity operator ##Px=-x## are then all eigenstates even or odd?
Yes, you're right, even or odd.LagrangeEuler said:But in the quantum linear harmonic oscillator case, you have even and odd eigenstates.
Sure, but since the eigenstates are non-degenerate they are also eigenstates of the parity operator and thus either even or odd.LagrangeEuler said:But in the quantum linear harmonic oscillator case, you have even and odd eigenstates.
What definition of even are you using? It's spherically symmetric, so ##\psi(\vec r) = \psi (-\vec r)##.LagrangeEuler said:Thank you. But just to understand. Ground state of hydrogen is
[tex]\psi_{100}(r)=C\mbox{e}^{-\frac{r}{a_0}}[/tex]
and this is not even neither odd function. Right?
This is even under parity. The parity operator acts on a wave function as ##\hat{P} \psi(\vec{x})=\psi(-\vec{x})##. Since for ##\ell=0## the wavefunction depends only on ##r=|\vec{x}|## it's automatically even under parity.LagrangeEuler said:Thank you. But just to understand. Ground state of hydrogen is
[tex]\psi_{100}(r)=C\mbox{e}^{-\frac{r}{a_0}}[/tex]
and this is not even neither odd function. Right?
A Hamiltonian is a mathematical operator used in quantum mechanics to describe the total energy of a system. It is represented by the symbol H and is a key component in the Schrödinger equation, which describes the time evolution of a quantum system.
A parity operator is a mathematical operator that describes the symmetry of a system under spatial inversion. It is represented by the symbol P and is used to determine whether a system is symmetric or asymmetric with respect to its spatial coordinates.
A Hamiltonian commutes with a parity operator if the two operators share the same set of eigenvectors, meaning that they have common eigenstates. This indicates that the Hamiltonian and the parity operator have compatible properties and can be used to describe the same physical system.
When a Hamiltonian commutes with a parity operator, it means that the Hamiltonian and the parity operator have the same set of eigenvectors and can be used interchangeably to describe the same physical system. This is useful in quantum mechanics because it allows for the simplification of calculations and the identification of important symmetries in a system.
It is important for a Hamiltonian to commute with a parity operator because it allows for the identification of important symmetries in a system, which can simplify calculations and provide insights into the behavior of the system. This is particularly useful in quantum mechanics, where understanding the symmetries of a system is crucial in predicting its behavior.