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Thursday, December 22, 2011

Quantum Mechanics (Uncertainity principle)

Question If a freely moving electron is localized in space to within $\Delta x_0$ of $x_0$, its wave function can be described by a wave packet $\psi(x,t)=\int_\infty^{-\infty}e^{i(kx-\omega t)}f(k)dk$, where $f(k)$ is peaked around a central value $k_0$. Which of the following is most nearly the width of the peak in $k$?
A. $\Delta k = 1/x_0$
B. $\Delta k = \frac{1}{\Delta x_0}$
C. $\Delta k = \frac{\Delta x_0}{x_0^2}$
D. $\Delta k = k_0\frac{\Delta x_0}{x_0}$
E. $\Delta k = \sqrt{k_0^2+(1/x_0)^2}$
Solution:
In quantum mechanics, the momentum $(p=\hbar{k})$ and position $(x)$ wave functions are Fourier transform pairs and the relation between $p$ and $x$ representations forms the Heisenberg uncertainty relation:
$\Delta{x}\Delta{k}\geq1 \Rightarrow \Delta k \geq \frac{1}{\Delta x}$
Answer: B

Symmetries and conservation Laws: Part 1


  • Every conservation principle corresponds to symmetry in nature
  • A symmetry of a particular kind exists when a certain operation leaves something unchanged.
  • There is an intimate connection between symmetry and so called conserved quantities.
  • Well known conserved quantity is energy and corresponding symmetry in this case is time translation.
Momentum Conservation
  • Holds for all type of interactions
  • Related to the invariance of physical laws under translation in space.
  • Thus laws of interaction do not depend on the place of measurement so the space is homogeneous.
  • This transnational uniformity of space leads to the conservation of linear momentum.
  • Particle at rest have no momentum. If it  decays into two less massive particles , momentum conservation requires that the two particles travel away in exactly opposite directions.
Conservation of Energy
  • Holds for all type of interactions.
  • related to the invariance of physical laws under translations along the time axis i.e., homogeneity of time.
  • laws of interaction do not depend on the time of measurement
Angular momentum conservation
  • In addition to transnational symmetry , space also has a rotational symmetry.
  • This symmetry of space gives rise to another conserved quantity , angular momentum.
  • This law is also of general validity for all types of interactions.
  • It is related to the invariance of the physical laws under rotation (isotropy of space).
  • The orbital and spin angular momentum may be separately conserved.
Parity Conservation
  • Holds for strong, nuclear and electromagnetic interactions but is violated in case of week interactions.
  • related to the invariance of the physical laws under inversion of space co-ordinates. x,y,z are replaced by -x.-y,-z.
  • is equivalent to combined reflection and rotation.
  • physical laws do not depend on the right handedness of co-ordinate system.
  • Parity operation symmetry represents discrete symmetry (reflection and rotation through 180 degree)
  • Every particle with non zero mass has an intrinsic parity  π which can either be +1(even) or -1 (odd). Thus total parity of a system of n particles is the product of their intrinsic parities and the orbital parity (-1)l.
  • Thus, πtot1π2π3.......πn(-1)l
  • Intrinsic parity of pions is odd.
Conservation of charge

  • Conservation of electric charge is related to gauge transformations which are shifts in the zeros of the scalar and vector electromagnetic potentials V and A
  • Gauge transformations leave E and B unaffected since the latter are obtained by differentiating potentials , and this invariance leads to charge conservation.
  • Charge and baryon number are conserved in all interactions.


Baryons

  • There is another whole class of unstable particles known as 'hyperons' , whose masses are each greater then that of protons.
  • The first hyperon was found in cosmic rays named as Λ0 hypron , a neutral decaying particle.
  • Charged particles seen in the decay were identified as proton and π- meson, indicating a process

    Λ0 →p+π-

  • Anti-Λ0 hypron decay to an anti proton and π+-meson.
  • The family of hyprons wit greatedt number of members is Σ-family.
  • First if it to be observed is Σ+ with mass about 2328me , and two prominent decat schemes are

    Σ+→p+π0 ; Σ+→n+π+

  • Σ- has just one set of decay products

    Σ-→n+π-

    its mass being slightly greater than Σ+ and is 2341 me.

  • Neutral Σ hyperon decays as

    Σ0→Λ0

    its mass is 2328me

  • The Σ hyperon form a triplet Σ+- and Σ0. A corresponding triplet of anti Σ hyperon also exists.
  • The anti particle of Σ+ can not be Σ- because the two have slightly different masses, whereas particle anti particle must have exactly same masses.
  • Another group of members belonging to hyperon family is Ξ hyperon originally called cascade particles.
  • Theit negative and neutral forms have been observed with decay processes

    Ξ-→Λ0-

    Ξ0→Λ00

    Their masses are about 2582 me

  • Anti Ξ hyperons have been detected.
  • Togather with nucleons (p and n) , the hyperons form the family of baryons.

Baryons(B=+1, Le=Lμ=Lτ=0)

Particle Symbol mean life (s) spin S Y I I3 Mass MeV/c2
Nucleon n stable 1/2 0 +1 1/2 -1/2 938.3
p 886 +1/2 936.6
Lambda Λ0 2.6×10-10 1/2 -1 0 0 0 1116
Sigma Σ+ 8.0×10-11 1/2 -1 0 1 +1 1189
Σ0 6×10-20 0 1193
Σ- 1.5×10-10 -1 1197
Xi Ξ0 2.9×10-10 1/2 -2 -1 1/2 +1/2 1315
Ξ- 1.6×10-10 -1/2 1321
Omega Ω- 8.2×10-11 3/2 -3 -2 0 1672
  • There is a sequence of decay for Ω- baryon

    Ω-→Ξ0-→Λ00 ....

    Final result of decay is proton , two electrons and two photons.

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