QM I - Decomposition of countable basic states into coherent states

In summary, the conversation discusses the calculation of a quantum system with a countable number of basic states and its decomposition into a basis of coherent states. The lowering and raising operators are also mentioned, but the specific approach and solution are still unclear.
  • #1
BasslineSanta
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Homework Statement



Consider a quantum system with a countable number of basic states [itex]\left|n\right\rangle[/itex].
Calculate the decomposition into a basis of coherent states [itex]\left|λ \right\rangle[/itex] all obeying [itex]\hat{a}[/itex] [itex]\left|λ \right\rangle[/itex] = λ [itex]\left|λ \right\rangle[/itex]


Homework Equations



[itex]\hat{a}[/itex] is the lowering operator:
[itex]\hat{a} \left|n\right\rangle[/itex] = √n [itex]\left|n-1\right\rangle[/itex]



The Attempt at a Solution



Because [itex]\left|λ\right\rangle[/itex] form a basis, i can equate [itex]\left|n\right\rangle[/itex] = Ʃλ[itex]_{n}\left|λ\right\rangle[/itex].
Applying the lowering operator n-times to both sides of the equation, i get: √n! [itex]\left|0\right\rangle[/itex] = λ[itex]^{n}[/itex] Ʃλ[itex]_{n}[/itex] [itex]\left|λ\right\rangle[/itex]
By equality of two vectors i can say that √n! = λ[itex]^{n}[/itex] and that [itex]\left|0\right\rangle[/itex] = Ʃλ[itex]_{n}[/itex] [itex]\left|λ\right\rangle[/itex].

Now i got kinda stuck. I thought if i get to the [itex]\left|0\right\rangle[/itex] vector, i can just keep applying the raising operator to get any state [itex]\left|n\right\rangle[/itex] written in my new vectors [itex]\left|λ\right\rangle[/itex]. But i realized i do not know how the raising operator acts on them. Neither do i know if i chose the right approach, but it feels like the only thing i could have done, considering the information given.

I would really appreciate some help.
Thanks a lot in advance!
 
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  • #2


I'm a little confused here as to whether you're trying to express a coherent state as a sum of the |n>'s, or one of the |n>'s as a sum of coherent states :confused:

The former is easy enough, but the latter is trickier - the answer won't be unique, because the coherent state basis is an overcomplete basis. Probably you'll have to use a coherent state resolution of unity, which involves integrating over the complex λ plane.
 

1. What is QM I - Decomposition of countable basic states into coherent states?

QM I - Decomposition of countable basic states into coherent states is a concept in quantum mechanics that describes how a system with a countable number of states can be broken down into coherent states, which are states with definite values for position and momentum. This technique is commonly used in quantum mechanics to simplify calculations and analyze the behavior of a system.

2. What are coherent states?

Coherent states are quantum states that have a well-defined value for both position and momentum. They are typically represented by a wavefunction that is a Gaussian function in both position and momentum space. These states are often described as the most classical-like states in quantum mechanics, as they exhibit properties similar to classical particles.

3. How are coherent states related to the Heisenberg uncertainty principle?

Coherent states are related to the Heisenberg uncertainty principle in that they represent a state of minimum uncertainty. This means that the product of the uncertainties in position and momentum for a coherent state is equal to the minimum allowed value, as dictated by the uncertainty principle. In other words, coherent states represent the most well-localized state possible.

4. What is the significance of decomposing countable basic states into coherent states?

Decomposing countable basic states into coherent states is significant because it allows for a simplification and a deeper understanding of quantum systems. By breaking down a system into coherent states, we can analyze the behavior of the system in a more classical-like manner and make predictions about its evolution over time. This technique is particularly useful in studying systems with a large number of states, such as the quantized harmonic oscillator.

5. How is the decomposition of countable basic states into coherent states performed?

The decomposition of countable basic states into coherent states is typically performed using a mathematical technique known as the resolution of identity. This involves expressing the identity operator in terms of coherent states, allowing us to write any state in the system as a linear combination of coherent states. This technique is similar to the Fourier transform in classical mechanics and is an essential tool in quantum mechanics for analyzing systems with a countable number of states.

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