Parity switching wave functions for a parity invariant hamiltonian?

In summary: Therefore, in even potentials like V(x) = λx^4, the eigenstates will occur with alternating parity, as stated by Shankar. In summary, Shankar discusses the Variational method for approximating wave functions and energy levels, using the example of an even potential V(x) = λx^4. He explains that due to the parity invariance of the Hamiltonian, the eigenstates will occur with alternating parity. This can also be understood mathematically through the number of nodes in each eigenfunction.
  • #1
VortexLattice
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Hi guys, I'm reading Shankar and he's talking about the Variational method for approximating wave functions and energy levels.

At one point he's using the example [itex]V(x) = λx^4[/itex], which is obviously an even function. He says "because H is parity invariant, the states will occur with alternating parity".

I believe him because I remember this from the Harmonic oscillator and the infinite square well, and I see why in each of those examples they alternate, from a mathematical standpoint.

But is there a better physical explanation? Right now all I can really tell is, mathematically, for those two examples, the solutions have to have alternating parity. So how can he say this for sure of all even potentials?

Thanks!
 
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  • #2
Anyone? I'm very curious.
 
  • #3
See Messiah, Vol I, Chapter III, "One-Dimensional Quantized Systems", Sect 12, "Number of Nodes of Bound States". If one arranges the eigenstates oin the order of increasing energies, the eigenfunctions likewise fall in the order of increasing number of nodes; the nth eigenfunction has (n-1) nodes between each of which the following eigenfunctions all have at least one node.

Regarding parity, each eigenfunction is either even or odd. If you have an even/odd number of nodes, you must have an even/odd eigenfunction.
 

FAQ: Parity switching wave functions for a parity invariant hamiltonian?

What are parity switching wave functions?

Parity switching wave functions refer to the wave functions of a quantum system that change sign under the operation of parity transformation. This means that the wave function changes sign when all spatial coordinates are inverted simultaneously. In other words, if the wave function is represented by ψ(x,y,z), then the parity transformed wave function would be -ψ(-x,-y,-z).

What is a parity invariant Hamiltonian?

A parity invariant Hamiltonian is a quantum mechanical Hamiltonian that remains unchanged under the operation of parity transformation. This means that the Hamiltonian operator commutes with the parity operator, and thus, the Hamiltonian is symmetric under parity transformation.

How are parity switching wave functions and parity invariant Hamiltonians related?

Parity switching wave functions and parity invariant Hamiltonians are related because for a system described by a parity invariant Hamiltonian, the wave functions must also be parity eigenstates. This means that the wave functions must either be even or odd under parity transformation, making them parity switching wave functions.

What is the significance of parity symmetry in quantum mechanics?

Parity symmetry is significant in quantum mechanics because it is a fundamental symmetry that helps us understand the behavior of physical systems. It is used to classify particles and their corresponding wave functions as either bosons (even parity) or fermions (odd parity), and it plays a crucial role in determining the selection rules for transitions between different energy states.

How can parity switching wave functions and parity invariant Hamiltonians be applied in practical situations?

Parity switching wave functions and parity invariant Hamiltonians have many practical applications in quantum mechanics, such as in the study of electronic and atomic structures, as well as in particle physics. They are also used in the development of quantum algorithms and quantum technologies, such as quantum computing and quantum cryptography.

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