Heisenberg’s uncertainty principle

ByMahantesh S NDept. of physics Manasagangothri Mysore

Contents

• Introduction

• Derivation of HUP

• Applications of UP

• Significance

• Remarks and

• references

o We have learned that the particles which occur in nature have wave properties. A moving particle with a well-defined

momentum p can behave like a wave of wavelength 𝝀 =𝒉

𝒑, and

this relation between momentum and wavelength is universal , i.e., valid for all real particles.

Introduction

o We have learned that the state of motion of a particle can be described by a complex wave function ψ x, t .

o For an isolated particle this wave function satisfies the Klein-Gordon equation, subject to the additional that only positive frequencies will occur in the Fourier resolution of the wave function.

o The initial wave function is quite arbitrary, and we can therefore have a very wide variety of different waves corresponding to different states of motion of the particle.

o It is important to understand that a wave in quantum mechanics need not look like a sine wave : that is a very special case.

o The Klein-Gordon equation determines the time dependence of the wave function , but it imposes no restriction on the “appearance” of the wave at some single instant of time.

o It will , however , restrict the appearance of the wave at two different times. The wave function ψ x, 𝑡1 at time t=𝑡1uniquely determines the state of motion of the particle.

o Consider now a state of motion of a particle described by the initial wave function ψ x, 0 . What can we say about the position and momentum of the particle at time t=0?

o The particle is most likely to be found in those regions in space in which the amplitude is large.

o If the initial wave function is such that the amplitude is zero except in a very small region , then we can say that the particle is in in the region: its position is accurately known.

o If its amplitude is approximately constant over a large region: the position at time t=0 is subjected to a large uncertainty.

o Analogous considerations apply to the momentum variable. Since momentum and wavelength are related by the de Broglie equation it is intuitively dear that the momentum cannot be well defined unless the wavelength is well defined.

o For the wavelength to be well defined it is necessary that the wave function exhibit some pattern of periodicity. precision to which momentum is defined depends on the state of motion of the particle: it can be very well defined or it can be very poorly defined.

Here comes the Heisenberg

He realized that whereas there are no limits to the accuracy to which either the momentum or the position can be defined , there is a fundamental limit to the accuracy to which position and momentum can be defined at the same time (i.e., for the same wave function).

This insight finds its expression in the celebrated “uncertainty relations” , formulated by German scientist Werner Heisenberg in 1927.

Δ𝑥 Δ𝑝𝑥 = ℎ/2𝜋

or

It states that “ It’s impossible to determine simultaneously

both momentum and position of the particle with accuracy”.

Illustration of uncertainty principle by Gamma ray microscope

Photon from the source when collide with electron, some photons enters into the microscope and enable the observer to find out the both the position an momentum of the electron. There are two limitations ,

1. Determination of the position of the electron is limited by the laws of optics. The limit of resolution of a microscope is

--------- (1)

If the position of the electron changes by a distance ∆𝑥 , the microscope is not able to detect it. To minimise ∆𝑥 , we must use radiation of shorter wavelength such as gamma rays. ∆𝑥 represents the uncertainty in position of the electron.

2. Use of gamma rays involved Compton effect. In order to observe the electron , one of the photons must strike the electron and then it should enter into the microscope anywhere between the angular range of –θ to +θ.

By de-Broglie relation 𝝀 =𝒉

𝒑,.it gives the momentum of

scattered photon. Its X-component is lies between ℎ

𝝀sinθ and −

ℎ

𝝀sinθ

∆𝑝𝑥 =ℎ

𝝀sinθ - ( −

ℎ

𝝀)sinθ

∆𝑝𝑥 = 2ℎ

𝝀sinθ ---------- (2)

This momentum is transferred to the electron by the scattered photon. The equation (2) gives the uncertainty in the measurement of momentum along the X-axis.

The product of these two equations gives ,

∆𝒙∆𝒑𝒙 = 𝒉

The more accurate value is 𝒉/𝟐𝝅 it is in good agreement with uncertainty principle.

APPLICATIONS OF UNCERTAINTY PRINCIPLE

According to the uncertainty principle it is not possible to measure simultaneously the position and momentum of the particle with the desired accuracy. The principle of uncertainty explains a large number of facts which could not be explained by classical mechanics. A few of them are,

Non-existence of free electrons in the nucleus.

Strength of the Nuclear forces.

Ground state Energy of H-atom and Radius of the Bohr’s first orbit.

Ground state Energy of harmonic oscillator.

Natural width of the spectral line.according to HUP of energy and time

𝜟𝑬 𝜟𝒕 ≥ ħ

since the lifetime of electron in an excited orbit is finite , therefore the

energy levels of the atom given by 𝜟𝑬= ħ/𝜟𝒕 must have the finite width. This means the excited energy levels of have a finite energy spread thereby indicating that the radiation emitted when an electron jumps truly monochromatic.

SOME REMARKS ABOUT UNCERTAINTY PRINCIPLESOME REMARKS ABOUT UNCERTAINTY PRINCIPLE

The discussion in the preceding sections proves that according to the UP , we can never obtain precise knowledge of a physical system, regardless of improvements in experimental techniques. We can only obtain probable values, and this is a characteristic of nature.

This principle provided support for the methods of wave mechanics. It shows that there was an uncertainty inherent in nature, and this was precisely what wave mechanics predicted.

ConclusionConclusion

Many phenomenon in the case of particles of atomic dimensions can be best understood in terms of uncertainty principle

Uncertainty principle is not applicable in our daily experiment. The calculation involving the measurement of both position and momentum of the terrestrial bodies remarkably accurate. Because the terrestrial bodies have appreciable mass , so the uncertainty in the measurement of position and momentum are negligible and may be neglected.

ReferencesReferences

Introduction to quantum mechanics by David J Griffiths.

Quantum mechanics principle and theory of measurement by Berkeley physics course by E H Wichman

Origin of Quantum Theory : Wave Mechanics Concept and Uncertainty principle

www.google.com and

Wikipedia.

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