# Ground State Wave function

• njoshi3

#### njoshi3

Hi,
Why is that, the ground state eigenfunction in ANY quantum mechanical system is symmetric under inversion of co-ordinates?

Originally posted by njoshi3
Hi,
Why is that, the ground state eigenfunction in ANY quantum mechanical system is symmetric under inversion of co-ordinates?

well first of all, this isn t true. the groundstate is only symmetric if the hamiltonian is also symmetric.

so let s assume that you had asked this question about any quantum system with a symmetric hamiltonian.

when there is a symmetric hamiltonian, all eigenfuctions must be either even or odd. that is, they must be either symmetric or antisymmetric.

an antisymmetric wavefunction necessarily passes through zero. therefore it has a node, therefore it cannot be the ground state.

therefore the ground state must be symmetric.

## 1. What is a ground state wave function?

A ground state wave function is a mathematical representation of the lowest energy state of a quantum mechanical system. It describes the probability of finding a particle in a certain location in space at a given time.

## 2. How is the ground state wave function different from an excited state wave function?

The ground state wave function has the lowest possible energy for a system, while excited state wave functions have higher energies. Additionally, the ground state wave function is a stationary state, meaning it does not change with time, while excited state wave functions do change with time.

## 3. Can the ground state wave function be negative?

Yes, the ground state wave function can have both positive and negative values. The square of the wave function, however, which represents the probability of finding a particle in a specific location, is always positive.

## 4. How is the ground state wave function related to the Schrödinger equation?

The ground state wave function is the solution to the time-independent Schrödinger equation, which describes the behavior of quantum systems. The Schrödinger equation allows us to calculate the ground state wave function for a given system.

## 5. How does the shape of the ground state wave function depend on the potential energy of the system?

The shape of the ground state wave function is directly related to the potential energy of the system. A higher potential energy will result in a narrower and taller wave function, while a lower potential energy will result in a wider and shorter wave function. The actual shape of the wave function is determined by the specific potential energy function of the system.