# 1D Groundstate wavefunction always even for even potential?

• Wavelet
In summary, the conversation discusses the ground-state wavefunction of a particle in a one-dimensional system with an even potential. It is stated that the ground-state wavefunction is always even and a proof for this is requested. The potential is an even function and it is believed that this is the reason for the evenness of the wavefunction. The conversation ends with a request for resources to find a rigorous proof for this fact.
Wavelet
Hi!
I have calculated various eigenstate wavefunctions for a one-dimensional system of a particle in a potential. The potential is an even function.
All the wavefunctions have become either even or odd functions which I understand why. The ground-state wavefunction has always been even, is this always the case and if so why? If not does anyone know of a system with odd ground-state wavefunction?

Wavelet

For a hamiltonian of the form H = p^2/2m + V(x) in one dimension, the ground state wave function has no nodes (that is, is never zero). So, if it must be odd or even (as it must if V is even), then it has to be even.

I don't have time now to type in a proof that the ground state wave function has no nodes, but it shouldn't be hard to find one on the web somewhere.

Thanks!

Hi Avodyne,

Good whatever time of day it is where you are.

I'm searching for a rigorous proof of the fact that for any even potential the ground state wave function is always even. Non-relativistic 1 dimensional case proof would be fine for me. Landau Lifschitz, volume #3 the next paragraph after (21.8) equation, produces a proof (if it can be called a proof), but it's not enough for example for V(x)=A*x^4-B*x^2 case.

So could you direct me in this search? A book or web-site?

With best regards,
Anton

## 1. What is a 1D groundstate wavefunction?

A 1D groundstate wavefunction is a mathematical representation of the lowest energy state of a particle in a one-dimensional system. It describes the probability amplitude of finding the particle at a certain position in space.

## 2. Why is the groundstate wavefunction always even for an even potential in 1D systems?

This is due to the symmetry of the even potential in 1D systems. The potential energy function is symmetric about the x-axis, meaning that the potential at -x is the same as the potential at x. This symmetry requires the wavefunction to also be symmetric about the x-axis, resulting in an even wavefunction.

## 3. Can the groundstate wavefunction be odd for an even potential in 1D systems?

No, the groundstate wavefunction must be even for an even potential in 1D systems. This is a consequence of the parity symmetry of the potential energy function.

## 4. How does the groundstate wavefunction change for an odd potential in 1D systems?

If the potential energy function is odd, meaning that it is not symmetric about the x-axis, then the groundstate wavefunction will also be odd. This is because the wavefunction must follow the symmetry of the potential energy function.

## 5. What is the significance of the groundstate wavefunction being even for even potential in 1D systems?

The evenness of the groundstate wavefunction for an even potential in 1D systems has important implications for the energy levels and properties of the system. It allows for the existence of bound states, where the particle is confined within a certain region of space, and affects the behavior of the system under perturbations or changes in the potential.

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