Time Dependent Perturbation Problem

In summary, the conversation involved a derivation of Ramsey Fringes using the Schrodinger equation and perturbation theory. The equations for the time dependence were found and an approximation technique was suggested to uncouple them. The result was a first order approximation of the wave function. The correctness of the calculations was confirmed.
  • #1
Diracobama2181
75
2
Homework Statement
Consider a two-level atom with the state space spanned by the two orthonormal states $$|1>$$ and $$|2>$$, dipole-coupled to an external time dependent driving field E(t). After a unitary transformation to the “rotating frame”, the Hamiltonian reads $$H/\hbar = ∆ |2><2| − f(t)[ |2><1|+|1><2|]$$, where ∆ is the difference between the atomic transition frequency ω0 and the frequency of the nearly monochromatic driving field ω, and f(t) is proportional to the temporal envelope of the driving field: E(t) ∝ f(t) cos(ωt). Suppose that $$f(t) =\frac{ λ }{2 \sqrt{\pi} \tau}(e^{-(\frac{ t + T/2}{\tau})^2}+e^{-(\frac{ t - T/2}{\tau})^2})$$ . This represents two pulses of light of length τ hitting the system at times ∓T/2. Let us assume that the amplitude of the driving field ∝ λ is “very small”, and that the system starts out in the state |1>.
Relevant Equations
$$i\hbar c_1(t)=<1|H'|2>c_2(t)$$
$$i\hbar c_2(t)=-f(t)c_2(t)$$
I am assuming this is the interaction picture, so I start with $$|\psi>=c_1(t)|1>+c_2(t)|2>$$. Plugging this into the Schrodinger equation,
I get the equations $$i\hbar c_1(t)=<1|H'|2>c_2(t)$$ and $$i\hbar c_2(t)=<1|H'|2>c_1(t)$$. I am assuming H' (the perturbation) is $$H'= − f(t)[ |2><2|+|1><1|]$$. From there, I get $$i\hbar c_1(t)=-f(t)c_2(t)$$, and $$i\hbar c_2(t)=-f(t)c_2(t)$$. Am I missing anything so far? I just want to make sure I haven't made any faults before continuing my calculation. I know this is supposed to be a derivation of Ramsey Fringes, it that is any help.
 
Physics news on Phys.org
  • #2
The time dependent equation should be :

##i\hbar dc_m (t)/dt= \sum_n H'_{mn}c_n(t)## where $$H'_{mn}=<m|H'|n>$$
 
  • #3
Note: Made a mistake writing the problem. It should be $$H'/\hbar = -f(t)[ |2><1|+|1><2|]$$.
 
  • #4
Abhishek11235 said:
The time dependent equation should be :

##i\hbar dc_m (t)/dt= \sum_n H'_{mn}c_n(t)## where $$H'_{mn}=<m|H'|n>$$
Using that equation and my correction, I get
$$ idc_1(t)/dt=-f(t)c_2(t)$$ and
$$ idc_2(t)/dt=-f(t)c_1(t)$$
Any idea on how I would uncouple these? Usually, I could take the second derivative of one and substitute, but the f(t) term makes that inapplicable.
 
  • #5
Diracobama2181 said:
Note: Made a mistake writing the problem. It should be $$H'/\hbar = -f(t)[ |2><1|+|1><2|]$$.
The diagonal term vanish and off diagonal terms survive with exponential factor(c.f Sakurai's Modern QM). You forgot to write :

##~i \hbar \mathbf{\dot c_1}= ...##
 
  • #6
Diracobama2181 said:
Using that equation and my correction, I get
$$ idc_1(t)/dt=-f(t)c_2(t)$$ and
$$ idc_2(t)/dt=-f(t)c_1(t)$$
Any idea on how I would uncouple these? Usually, I could take the second derivative of one and substitute, but the f(t) term makes that inapplicable.
Try using Dyson Series or any other approximation technique
 
  • #7
Ok, using a first order approximation, I get
$$c_1(t)=-\frac{i}{\hbar}\int_{-\infty}^{0}<1|(-\hbar f (t))(|2><1|+|1><2|)|2>dt$$
and
$$c_2(t)=-\frac{i}{\hbar}\int_{-\infty}^{0}<2|(-\hbar f(t))(|2><1|+|1><2|)|1>dt$$

or

$$c_1(t)=-\frac{i}{\hbar}\int_{-\infty}^{t}-\hbar f(t)dt$$
and
$$c_2(t)=-\frac{i}{\hbar}\int_{-\infty}^{t}-\hbar f(t)dt$$

Hence,
since $$C_1=1$$ and $$C_2=0$$ to zeroth order,

$$|\psi>=(1+c_1)|1>+c_2|2>$$.

Does this seem correct thus far?
 
  • #8
Diracobama2181 said:
Ok, using a first order approximation, I get
$$c_1(t)=-\frac{i}{\hbar}\int_{-\infty}^{0}<1|(-\hbar f (t))(|2><1|+|1><2|)|2>dt$$
and
$$c_2(t)=-\frac{i}{\hbar}\int_{-\infty}^{0}<2|(-\hbar f(t))(|2><1|+|1><2|)|1>dt$$

or

$$c_1(t)=-\frac{i}{\hbar}\int_{-\infty}^{t}-\hbar f(t)dt$$
and
$$c_2(t)=-\frac{i}{\hbar}\int_{-\infty}^{t}-\hbar f(t)dt$$

Hence,
since $$C_1=1$$ and $$C_2=0$$ to zeroth order,

$$|\psi>=(1+c_1)|1>+c_2|2>$$.

Does this seem correct thus far?

Seems alright to me
If you have done correctly, then it's all right(I didn't check equation or integration)
 

Related to Time Dependent Perturbation Problem

What is the time dependent perturbation problem?

The time dependent perturbation problem is a mathematical model used in quantum mechanics to study the behavior of a system when subjected to a perturbation, or a small change, over time. It helps to understand how the system's energy levels and wave functions are affected by the perturbation.

What are some examples of time dependent perturbations?

Some common examples of time dependent perturbations include an external electric or magnetic field, a change in temperature, or a sudden change in the system's boundary conditions. These perturbations can cause the system to deviate from its original state and exhibit different behaviors.

How is the time dependent perturbation problem solved?

The time dependent perturbation problem is typically solved using perturbation theory, which involves breaking down the problem into simpler, solvable parts. The solution is then obtained by summing up the contributions from each part. In some cases, numerical methods may also be used to solve the problem.

What is the significance of the time dependent perturbation problem?

The time dependent perturbation problem is a fundamental concept in quantum mechanics and is used to understand the behavior of physical systems at the atomic and subatomic level. It also has practical applications in fields such as chemistry, materials science, and engineering.

What are some challenges associated with the time dependent perturbation problem?

One of the main challenges of the time dependent perturbation problem is finding an accurate and efficient way to solve it. In some cases, the perturbation may be too large for perturbation theory to be applicable, or the system may be too complex to solve using traditional methods. In these cases, alternative approaches such as computational methods may be used.

Similar threads

  • Advanced Physics Homework Help
Replies
7
Views
969
  • Advanced Physics Homework Help
Replies
3
Views
831
  • Advanced Physics Homework Help
Replies
13
Views
2K
  • Advanced Physics Homework Help
2
Replies
36
Views
3K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
6
Views
557
  • Advanced Physics Homework Help
Replies
0
Views
519
  • Calculus and Beyond Homework Help
Replies
8
Views
383
  • Advanced Physics Homework Help
Replies
4
Views
1K
  • Advanced Physics Homework Help
Replies
13
Views
1K
Back
Top