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, π_tot=π₁π₂π₃.......π_n(-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 2328m_e , 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 m_e.

• Neutral Σ hyperon decays as

  Σ⁰→Λ⁰+γ

  its mass is 2328m_e

• The Σ hyperon form a triplet Σ⁺,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+π^-

  Ξ⁰→Λ⁰+π⁰

  Their masses are about 2582 m_e

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

Baryons(B=+1, L_e=L_μ=L_τ=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+π^-→Λ⁰ +π⁰ ....

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