Physics Assignment: Investigating Particle Behavior in Magnetic Fields

This assignment explores the dynamics of a particle, specifically an electron, within a magnetic field. It begins by examining the properties of the x operator and its implications for the system's behavior. The core of the assignment focuses on the time evolution of a particle in a magnetic field, considering its spin and the Hamiltonian governing the system. The solution describes the system's behavior, particularly when the magnetic field's time-dependent component points along the x-axis and rotates with an angular velocity. The assignment utilizes concepts from quantum mechanics, including the Hamiltonian and time evolution operators, to analyze the particle's motion and the influence of the magnetic field. The assignment also references key literature on the topic.

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

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 ≡ ¿ e∨β
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.

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.