Heisenberg principle free#

For the Gaussian function, the uncertainty product is To solve this problem, we must be specific about what is meant by “uncertainty of position” and “uncertainty of momentum.” We identify the uncertainty of position ( Δ x ) ( Δ x ) with the standard deviation of position ( σ x ) ( σ x ), and the uncertainty of momentum ( Δ p ) ( Δ p ) with the standard deviation of momentum ( σ p ) ( σ p ). This average energy of a particle is related to its average of the momentum squared, which is related to its momentum uncertainty. We can take the average energy of a particle described by this function ( E) as a good estimate of the ground state energy ( E 0 ) ( E 0 ).

Note that this function is very similar in shape to a Gaussian (bell curve) function. This is the largest wavelength that can “fit” in the box, so the wave function corresponds to the lowest energy state. The ground-state wave function of this system is a half wave, like that given in Example 7.1. ( Hint: According to early experiments, the size of a hydrogen atom is approximately 0.1 nm.)Īn electron bound to a hydrogen atom can be modeled by a particle bound to a one-dimensional box of length L = 0.1 nm. According to Heisenberg, these uncertainties obey the following relation.Įstimate the ground-state energy of a hydrogen atom using Heisenberg’s uncertainty principle. The particle can be better localized ( Δ x ( Δ x can be decreased) if more plane-wave states of different wavelengths or momenta are added together in the right way ( Δ p ( Δ p is increased). A wave packet contains many wavelengths and therefore by de Broglie’s relations many momenta-possible in quantum mechanics! This particle also has many values of position, although the particle is confined mostly to the interval Δ x Δ x. An example of a wave packet is shown in Figure 7.9. In quantum theory, a localized particle is modeled by a linear superposition of free-particle (or plane-wave) states called a wave packet. Similar statements can be made of localized particles.

Heisenberg principle free#

This account of a free particle is consistent with Heisenberg’s uncertainty principle. The uncertainty of position is infinite (we are completely uncertain about position) and the uncertainty of the momentum is zero (we are completely certain about momentum). The particle is equally likely to be found anywhere along the x-axis but has definite values of wavelength and wave number, and therefore momentum. As discussed in the previous section, the wave function for this particle is given byĪnd the probability density | ψ k ( x, t ) | 2 = A 2 | ψ k ( x, t ) | 2 = A 2 is uniform and independent of time. According to de Broglie’s relations, p = ℏ k p = ℏ k and E = ℏ ω E = ℏ ω. The particle moves with a constant velocity u and momentum p = m u p = m u. To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x-direction. We discuss the momentum-position and energy-time uncertainty principles separately.

Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Heisenberg’s uncertainty principle is a key principle in quantum mechanics. Describe the physical meaning of the energy-time uncertainty relation.

Explain the origins of the uncertainty principle in quantum theory.

Describe the physical meaning of the position-momentum uncertainty relation.

By the end of this section, you will be able to: