Question 1. Show that. Solution. Using the logic of Sak

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The Assignment has been carefully translated to English (handwritten). The solution can be handwritten also, in fact i prefer it better. No need for external citations (unless really necessary). Any questions please feel free to Email.

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Question 1
Show that ¿ p'
|x|α >¿ i
p' < p 'α >¿
Solution
Using the logic of Sakurai and having x as the generator of boosts (Patrick Norman, 2018);
x=i lim
p 0
1
p (κ ( p )κ ( 0 ) )
Where κ ( 0 ) =id
When x act on a state ¿ α >¿:
xα >¿i lim
p 0
1
p ¿ ¿
Contract with ¿ p>¿ to get
¿ p|x| α >¿i lim
p 0
¿ pΔ p|α >¿ p |α > ¿
p ¿
¿ i
p' < p ' α >¿
Show that ¿ β |x|α = d p' β ( p' ) i
p' α ( p' )
Solution
Consider the matrix elements of x
¿ β | x|α >¿d p' < β p'>¿ p 'xα>¿ ¿CITATION Pop 11¿ 2057 (Paul , 2011)
¿d p' < β | p' >i
p' < p'
| α >¿ ¿
¿d p' β ( p' ) i
p' α ( p' )
What is the physical significance of the operator Z=exp (ixγ
)

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Where x is position operator and γ is the number with the same dimension as momentum.
It shows that the periodicity of the operator yields a periodic motion of the oscillator. The
operator involves the momentum generated shifts in position.
Question 2
Particle in magnetic field. An electron with spin (S=1
2 ) is an external magnetic field in z-
direction Bz. The Hamiltonian is H=( e B
mc ) Sz=ω Sz where ω ¿ eβ
mc
Describe the dynamics of this system at t=0 and in a ¿ SX ,+¿ state.
Solution
At t=0, the time dependent part of the field points along the horizontal axis(x-axis) and is
rotating with an angular velocity ω in a clockwise direction (Walter Dittrich, 2012). The spin
Hamiltonian is
H ( t ) =γB Sz
Where B= e
mc
For ¿ Sx , t> ¿, we have a unitary operator B associated with the H that evolves ¿ Sx>¿ in time.
¿ Sx , t> ¿ is simply the time evolution from the t=0 state generated by H.
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References
Patrick Norman, K. R. T. S., 2018. Principles and Practices of Molecular Properties: Theory, Modeling, and
Simulations. In: s.l.:John Wiley & Sons, pp. 112-118.
Paul, P., 2011. Solving The Schrodinger Equation: Has Everything Been Tried?. In: s.l.:World Scientific, p.
132.
Walter Dittrich, M. R., 2012. Classical and Quantum Dynamics: from Classical Paths to Path Integrals. In:
s.l.:Springer Science & Business Media, pp. 67-69.
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