QM: Writing time evolution as sum over energy eigenstates

In summary, the conversation discusses a 1-D harmonic oscillator with angular velocity ##\omega## and eigenstates ##|j>##, where the state at ##t=0## is given by ##|\Psi(0)>##. The formula ##\hat{U}(t) = \sum_j e^{-iE_jt/\hbar}|j\rangle\langle j|## is then suggested as a way to write ##\hat{U}(t)## as a sum over energy eigenstates. The speaker is seeking help with the second part of the exercise, which involves using this formula.
  • #1
Muizz
11
0
Suppose I have a 1-D harmonic oscilator with angular velocity ##\omega## and eigenstates ##|j>## and let the state at ##t=0## be given by ##|\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates.

I've previously shown that ##\hat{H} = \sum_j |j>E_j<j|## with ##E_j = \hbar\omega(\frac12+j)## (part one of the exercise I got this from) but with this second part I'm drawing a complete blank. Any help would be hugely appreciated.
 
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  • #2
Hello,

Make life simple and imagine ##|\Psi(0)\rangle = |j\rangle ##.
What would ##\hat U(t) ## look like in that case ?
 

FAQ: QM: Writing time evolution as sum over energy eigenstates

What is the concept of time evolution in quantum mechanics?

Time evolution in quantum mechanics refers to the mathematical description of how a quantum system changes over time. It is based on the Schrödinger equation, which describes the evolution of the state of a quantum system in terms of the Hamiltonian operator.

What is meant by "sum over energy eigenstates" in relation to time evolution in quantum mechanics?

Sum over energy eigenstates refers to the decomposition of the state of a quantum system into a sum of its energy eigenstates. This allows us to express the time evolution of the system as a linear combination of these eigenstates, making it easier to calculate and understand the behavior of the system over time.

How do we write the time evolution of a quantum system as a sum over energy eigenstates?

The time evolution of a quantum system can be written as a sum over energy eigenstates by expressing the state of the system at any given time as a linear combination of the energy eigenstates. This involves using the Schrödinger equation and the Hamiltonian operator to calculate the coefficients of the eigenstates in the linear combination.

What are the benefits of writing time evolution as a sum over energy eigenstates?

Writing time evolution as a sum over energy eigenstates allows us to easily calculate the behavior of a quantum system over time, as well as gain insight into the energy levels of the system. It also simplifies the calculations and makes it easier to understand the evolution of the system.

Can the concept of time evolution as a sum over energy eigenstates be applied to all quantum systems?

Yes, the concept of time evolution as a sum over energy eigenstates applies to all quantum systems, as it is based on the fundamental principles of quantum mechanics. However, the specific calculations and eigenstates may differ depending on the system being studied.

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