Some Studies On Exact Solutions Of Models In Noncommutative Spaces

Abstract

The central theme of my thesis is to explore various simple prototype models that are exactly solvable in the framework of time dependent noncommutative spaces. By adopting the methodology provided by the Lewis Riesenfeld theory, we developed a procedure for obtaining a class of exact solutions for such model systems. We analyzed these solutions by deriving the energy expectation values analytically and representing those energy dynamics graphically. We also examined the explicit existence of a non-zero Berry geometric phase in the noncommutative framework and analyzed the role of noncommutativity in generating a non-trivial Berry phase when the model Hamiltonian and the noncommutative parameters are periodic in time. Overall, my thesis contributes to a deeper understanding of quantum theory in time dependent noncommutative backgrounds and indicates a strong possibility for developing a consistent quantum theory within such frameworks.

Figures (8)

• Figure 1: Graph of the exponential energy expectation value (divided by $\omega_0$ to eliminate the dimension), with respect to $\Gamma$t (also a dimensionless parameter). The numerical value (in natural units) of the constants are chosen as $M,\,\Gamma,\,\mu\,=\,1$; $\sigma,\,\Delta$=$10^7$; $\omega_0$=$10^3$ and the quantum state is fixed as $m=0, n=1$. The energy expectation value $\langle E \rangle$ is computed when $\langle A\rangle$ Set-IA : $\omega(t)=\omega_0 e^{-{\Gamma}t/2}$ and $f(t)=1$ ; $\langle B\rangle$ Set-IB : $\omega(t)=\omega_0$ and $f(t)=e^{-{\Gamma}t}$ ; $\langle C\rangle$ Set-IC : $\omega(t)=\omega_0 e^{-{\Gamma}t/2}$ and $f(t)=e^{-{\Gamma}t}$. For $\langle A\rangle$, the energy initially decays and then starts to grow with respect to time. For $\langle B\rangle$, the energy is always constant over time. For $\langle C\rangle$, the energy decreases and tends to zero over time.
• Figure 2: Graph of the rational energy expectation value (divided by $\omega_0$ to eliminate the dimension), with respect to $\Gamma$t (also a dimensionless parameter). The numerical value (in natural units) of the constants are chosen as $M,\,\Gamma,\,\mu\,\chi\,=\,1$; $\sigma,\,\Delta$=$10^7$; $\omega_0$=$10^3$ and the quantum state is fixed as $n=1, m=0$. The expectation value of energy $\langle E \rangle$ is computed when $\omega(t)=\dfrac{\omega_0}{(\Gamma\,t+\chi)}$ and $f(t)=1$.
• Figure 3: Graph of the energy expectation value (divided by $\omega_0$ to eliminate the dimension) corresponding to the elementary EP solutions, with respect to $\Gamma$t (also a dimensionless parameter). The numerical value (in natural units) of the constants are chosen as $M,\,\Gamma,\,\mu\,\chi\,=\,1$; $\sigma,\,\Delta$=$10^7$; $\omega_0$=$10^3$ and the quantum state is fixed as $m=0, n=1$. The energy expectation value $\langle E \rangle$ is computed when $\omega(t)=\dfrac{\omega_0}{(\Gamma\,t+\chi)}$ and $f(t)=1$.
• Figure 4: Graph of the exponential energy expectation value (divided by $\omega_0$ to eliminate the dimension), with respect to $\Gamma$t (also a dimensionless parameter). The numerical value (in natural units) of the constants are chosen as $M, q, \mu, \Gamma\,=\,1$, $B_0\,=\,10^2$, $\omega_0\,=\,10^3$, $\Delta, \sigma\,=\,10^7$ and the quantum state is fixed as $n=1, m=0$. The expectation value of energy $\langle E \rangle$ is computed for case I: $B(t)=B_0$, $\omega(t)=\omega_0$ and $f(t)=e^{-\Gamma\,t}$; case II: $B(t)=B_0\,e^{\Gamma\,t}$, $\omega(t)=\omega_0$ and $f(t)=e^{-\Gamma\,t}$; case III: $\omega(t)=\omega_0$, $B(t)=B_0e^{-\Gamma\,t}$ and $f(t)=e^{-\Gamma\,t}$, case IV: $B(t)=B_0e^{\Gamma\,t}$, $\omega(t)=\omega_0e^{-\Gamma\,t/2}$ and $f(t)=e^{-\Gamma\,t}$. For cases I and III, the energy starts to decay at first but ultimately becomes constant over time. For case II, the energy always stays constant over time. However, for case IV, the dynamics of energy are observed to be very interesting. Although it decays at first, it then starts to grow over time. The corresponding scenarios in the absence of the magnetic field are also plotted for cases I and IV for comparison purposes. For the cases where the angular frequency is set to be constant (cases I, II, and III), the energy expectation value also becomes constant when the external field is switched off. Thus, the external field, both in time varying and constant forms, influences the time evolution of the energy expectation value of a damped oscillator with a constant frequency. However, for case IV, where the angular frequency is a decaying function, the energy does not exhibit its expanding nature after removing the external field, which is an expanding function with time. Next, in case I, we remove damping by considering $f(t)=1$. This also results in a constant energy value. This energetics actually match those in case II, where the damping effect is nullified in the presence of an expanding magnetic field.
• Figure 5: Graph of the rational energy expectation value (divided by $\omega_0$ to eliminate the dimension), with respect to $\Gamma$t (also a dimensionless parameter). The numerical value (in natural units) of the constants are chosen as $M, q, \mu, \Gamma\,=\,1$, $B_0\,=\,10^{20}$, $\omega_0\,=\,10^3$, $\Delta, \sigma\,=\,10^7$ and the quantum state is fixed as $n=1, m=0$. The expectation value of energy $\langle E \rangle$ is computed for case I: $a(t)=\dfrac{4\sigma}{(\Gamma{t}+\chi)^{\,2}}, \,\rho(t)=\left[\dfrac{2\mu^{\,2}}{\Gamma{t}+\chi}\right]^{1/2},\,b(t)\,=\,\Delta\,$ and $\langle B\rangle$ case II: $a=\sigma ~~,~~ \rho=\mu \sqrt{\Gamma\,t+\chi}~~,~~ b(t)=\dfrac{\Delta}{(\Gamma\,t+\chi)^2}$. Although the energy in both cases decays rationally over time, the decay rate in case I is greater than in the other case. This is because, in case I, both parameters $a(t)$ and $\rho(t)$ are decaying functions, while in case II, only $b(t)$ is a decaying function and $\rho(t)$ increases. The corresponding scenarios in the absence of the magnetic field are plotted for both cases. The oscillator's energy also decays when the external field is switched off. Although the nature of the energy decay remains the same in both the presence and absence of the field, the energetics increase in value when the field is applied.