The Heisenberg Uncertainty Princeple

[FlyInDance]

Hey Guys, I've got a question, could you help me?
Can you tell me why the minimum of the Heisenberg Uncertainty Princeple is h bar over two, and how can we say the associate state is the Guass state.
I hope you can do it more mathematically.

 

[wavemaster]

See 3.3 and 3.4.3 in Griffiths.

 

[denian]

The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics that states that it is impossible to know the exact position and momentum of a particle simultaneously. This is due to the inherent uncertainty and randomness in the behavior of particles at the quantum level.

The minimum value of the uncertainty principle, or the minimum uncertainty in position and momentum, is represented by the symbol h bar over two. This value is derived from the mathematical formulation of the uncertainty principle, which involves the Planck constant (h) and the reduced Planck constant (h bar), and is a fundamental constant in quantum mechanics.

The associated state in the Heisenberg Uncertainty Principle is known as the Gaussian state, which refers to a state of a particle with a wavefunction that follows a Gaussian distribution. This state is often used in quantum mechanics as it simplifies the mathematical calculations involved in solving for the uncertainty principle.

To understand the association between the Heisenberg Uncertainty Principle and the Gaussian state mathematically, we can look at the wavefunction of a Gaussian state, which is a solution to the Schrödinger equation. The Gaussian wavefunction has a well-defined position and momentum, and its probability distribution has a minimum uncertainty, making it an ideal state to represent the uncertainty principle.

In summary, the minimum value of the Heisenberg Uncertainty Principle is h bar over two, and the associated state is the Gaussian state, which represents the minimum uncertainty in position and momentum. This is a fundamental principle in quantum mechanics and plays a crucial role in understanding the behavior of particles at the quantum level.