One-dimensional bound states are a type of quantum mechanical system where a **particle is confined to move in one dimension** and is unable to escape a particular region of space.

## Properties of One-Dimensional Bound States

## #1. Energy quantization:

In a one-dimensional bound state, the energy of the particle is quantized.

This means that the particle can only have certain discrete values of energy and cannot have any value in between.

## #2. Finite potential well:

In a one-dimensional bound state, the particle is typically confined by a potential well.

The potential well has finite depth, which means that the particle can tunnel through the potential barrier, but the probability of doing so decreases exponentially with increasing barrier height.

## #3. Degeneracy:

In some cases, the one-dimensional bound state may have degenerate energy levels, meaning that two or more states have the same energy.

This can occur when the potential well has a particular symmetry.

## #4. Normalization:

The wavefunction of the particle must be normalized, which means that the total probability of finding the particle in the region of confinement must be equal to one.

## #5. Nodes:

The wavefunction may have nodes, which are points where the probability of finding the particle is zero.

The number of nodes depends on the energy level of the particle,

Example,

- The ground state (n=1) wavefunction has no nodes, meaning that the probability of finding the particle is non-zero everywhere within the region of confinement.
- The first excited state (n=2) wavefunction has one node, the second excited state (n=3) wavefunction has two nodes, and so on.

In general, the nth excited state wavefunction has n-1 nodes.

**Position of Nodes**

The positions of the nodes depend on the specific potential well confining the particle, and the solutions to the SchrÃ¶dinger equation for the system can be used to calculate the positions of the nodes for each energy level.