De-Broglie Wavelength:
The de-Broglie wavelength, denoted by λ, is a concept in quantum mechanics that relates the momentum of a particle to its wavelength. It states that every object, whether it is a particle or a wave, has a wavelength associated with it.

Uncertainty Principle:
The uncertainty principle, proposed by Werner Heisenberg, is a fundamental principle of quantum mechanics. It states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle can be known simultaneously. In particular, it states that the more precisely one property is known, the less precisely the other can be known.

Uncertainty Principle for a Free Particle in One Dimension:
The uncertainty principle for a free particle in one dimension can be expressed mathematically as:

Δx * Δp ≥ ħ/2

where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant (h/2π).

Explanation:
1. Position Uncertainty:
The uncertainty in position, Δx, refers to how precisely we can determine the position of a particle along the x-axis. It represents the range of possible positions that the particle could occupy.

2. Momentum Uncertainty:
The uncertainty in momentum, Δp, refers to how precisely we can determine the momentum of a particle along the x-axis. It represents the range of possible momenta that the particle could have.

3. Product of Uncertainties:
According to the uncertainty principle, the product of the uncertainties in position and momentum must be greater than or equal to the reduced Planck's constant divided by 2.

4. Physical Interpretation:
The uncertainty principle implies that it is impossible to simultaneously know both the exact position and exact momentum of a particle. This is not due to any limitations in measurement techniques, but rather a fundamental property of quantum mechanics.

5. Wave-Particle Duality:
The uncertainty principle is closely related to the wave-particle duality of quantum mechanics. It suggests that particles can exhibit both particle-like and wave-like properties. The de-Broglie wavelength is a manifestation of this wave-particle duality, as it relates the momentum of a particle to its wavelength.

6. Implications:
The uncertainty principle has profound implications for our understanding of the microscopic world. It places limitations on the precision with which certain physical properties can be known, and introduces a fundamental level of randomness and unpredictability into quantum systems.

7. Applications:
The uncertainty principle has numerous applications in various fields of physics, such as quantum mechanics, quantum field theory, and quantum computing. It is a fundamental principle that underlies many phenomena and experimental results in the realm of quantum physics.

In summary, the uncertainty principle for a free particle in one dimension states that there is a fundamental limit to the precision with which the position and momentum of a particle can be known simultaneously. The de-Broglie wavelength is a concept that relates the momentum of a particle to its wavelength, and is closely connected to the wave-particle duality of quantum mechanics.