# Solving the Uncertainty Principle: Kinetic Energy & Position

• crysien
In summary: Therefore, the uncertainty principle still holds and the product of the uncertainties in KE and position is greater than or equal to h/(4π). In summary, the uncertainty relation for kinetic energy and position in a 1-D particle in a box is given by the equation h/(4π) >= ΔKE Δx, which does not imply that the actual uncertainties are zero, but rather that their product must be greater than or equal to h/(4π).
crysien
I am a little confused by something by something in my physical chemistry textbook.

If two measurable quantities do not commute, then an uncertainty relation exists for them. Kinetic energy and position do not commute, and the expectation value for linear momentum in a 1-D particle in a box is zero. However, the uncertainty relation for KE and position says that:

Obviously, the uncertainty can't be zero, but I don't see why this equation is correct and I haven't found anything working out the actual integral to reach this result online. Could someone please explain?

The uncertainty relation for kinetic energy and position states that the product of the uncertainties in these two quantities must be greater than or equal to a certain value. This value is known as the "uncertainty principle" and is equal to h/(4π), where h is Planck's constant. The equation you have given is simply a statement of this principle; it does not imply that the actual values for the uncertainties in these two quantities are zero. In fact, the equations for linear momentum in a 1-D particle in a box only imply that the expectation value for linear momentum is zero, not necessarily the actual value.

## 1. What is the Uncertainty Principle?

The Uncertainty Principle, also known as Heisenberg's Uncertainty Principle, states that it is impossible to know both the exact position and exact momentum of a particle simultaneously.

## 2. How does the Uncertainty Principle relate to Kinetic Energy and Position?

The Uncertainty Principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. This means that as we try to measure the position of a particle more accurately, we introduce more uncertainty in its momentum, which is directly related to its kinetic energy.

## 3. Can the Uncertainty Principle be violated?

No, the Uncertainty Principle is a fundamental principle in quantum mechanics and has been extensively tested and proven to be true. It is a fundamental limitation of our ability to measure and understand the behavior of particles at the quantum level.

## 4. How does solving the Uncertainty Principle benefit science?

Solving the Uncertainty Principle allows us to better understand the behavior of particles at the quantum level, which is crucial in fields such as quantum mechanics, atomic and molecular physics, and nanotechnology. It also helps us improve our measurement techniques and develop new technologies.

## 5. Are there any applications of the Uncertainty Principle in everyday life?

While the Uncertainty Principle may seem abstract and theoretical, its applications can be seen in everyday life through technologies such as MRI machines, which use the principles of quantum mechanics to create detailed images of the human body. It also plays a role in the development of new materials and technologies in fields such as computer science and communication.

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