Time-evolution of a quantum system

In summary, we are considering an electron bound in a hydrogen atom under the influence of a homogeneous magnetic field. The Hamiltonian of the system is given by H = H0 - ωLz, where H0 is the Hamiltonian of the unperturbed hydrogen atom and ω is a constant dependent on the magnetic field. At t = 0, the system is in the state |ψ(t=0)> = 1/\sqrt{2}(|2,1,-1> - |2,1,1>). We are asked to calculate the probabilities of finding the system in three different states at some later time t > 0. To do this, we use the time evolution equation |ψ(t)> =
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Homework Statement


Consider an electron bound in a hydrogen atom under the influence of a homogenous
magnetic field B = zˆB  . Ignore the electron spin. The Hamiltonian of the system is H = H0 −ωLz ,where
H0 is the Hamiltonian of the hydrogen atom with the usual eigenstates nlm and eigenenergies (0) En
(we use the superscript (0) to denote the unperturbed hydrogen atom), and ω =|e |B/(2mec).

At t = 0 the system is in the state: |ψ(t=0)>=1/[itex]\sqrt{2}([/itex](|2,1,-1> -|2,1,1>)
For each of the following states
calculate the probability of finding the system at some later time t > 0 in that state:

1(t)>=1/[itex]\sqrt{2}(|2,1,-1> - |2,1,1>)

2(t)>=1/[itex]\sqrt{2}(|2,1,-1> + |2,1,1>)

3(t)>=1/[itex]\sqrt{2}(|2,1,0>

Homework Equations



|ψ(t)>=U(t)*|ψ(t=0)>
U(t)=exp(-i*H*t/\hbar)

The Attempt at a Solution



So the time evolution of a system can be described by multiplying the wavefunction at t=0 by the function U(t) which in this case would be exp(-i*(En(o)-|e |B/(2mec)*t/\hbar). But then from there, I'm a bit confused. Do I need to transform the equation into a matrix form and then take the square of the coefficients of the superimposed wavefunctions to get the probabilities? If that was the case, then I'd be able to represent |ψ1(t)>=1/\sqrt{2}[-1,0,1], |ψ2(t)>=1/\sqrt{2}[1,0,1] and then |ψ3(t)>=[0,1,0] right?

Thanks for your help.
 
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  • #2
Ok, so I tried the matrix approach, but it doesn't seem quite right. Here is what I did:

With U(t)=exp(-i*(En(o)-|e |B/(2mec)*t/\hbar) I split that up into
U(t)=exp(-i*En(o)t/hbar)*exp(i*α*t/\hbar) where α=|e|B/(2mec)

from there I used the equation exp(iθ[itex]\widehat{n}[/itex][itex]\bullet[/itex][itex]\vec{σ}[/itex])=Icos(θ)+i[itex]\widehat{n}[/itex][itex]\bullet[/itex][itex]\vec{σ}[/itex]sin(θ)

where 'I' is the identity matrix

which gave me Icos(αt)+iBLzsin(αt) where I used the matrix representation of Lz

Now multiplying this new expression for U(t) gives me
|ψ(t)>=exp(i*En(o)*(cos(αt)|ψ1> -i*B*hbar*sin(αt)|ψ2>)

And we can take the square of the coefficients of the wavefunctions as their probabilities. So for example, the probability of |ψ3> would be 0, because it doesn't appear in the equation. Now, apart from appear really tenuous, the squares of the probabilities don't add up to 1, which makes me really doubt this is correct. Maybe I'd need to integrate from t=0 --> infinity, but that's a rather hard integral, and not likely either. Am I at least close?
 

FAQ: Time-evolution of a quantum system

1. What is time-evolution of a quantum system?

The time-evolution of a quantum system refers to the way in which the state of a quantum system changes over time. This is governed by the Schrödinger equation, which describes how the wave function of a quantum system evolves.

2. How does time-evolution differ in classical and quantum systems?

In classical systems, time-evolution is described by deterministic equations such as Newton's laws of motion. However, in quantum systems, time-evolution is probabilistic and described by the Schrödinger equation. This means that the state of a quantum system is not completely predictable, but rather can be described by a probability distribution.

3. Can the time-evolution of a quantum system be reversed?

No, the time-evolution of a quantum system is irreversible. This is due to the probabilistic nature of quantum mechanics, which means that the exact initial state of a system cannot be determined from its final state.

4. What factors affect the time-evolution of a quantum system?

The time-evolution of a quantum system is influenced by a number of factors, including the initial state of the system, the Hamiltonian (or energy operator) of the system, and any external forces or interactions acting on the system.

5. How is time-evolution observed in experiments?

Time-evolution of a quantum system can be observed indirectly through measurements of physical quantities, such as energy or position, that are affected by the system's state. These measurements can be used to determine the probability distribution of the system's state at different points in time.

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