Show that a complex PE yields a time dependant probability of finding a particle

In summary, the conversation discusses how to show that a complex potential energy leads to an unstable particle and difficulty in ensuring normalization of probability over time. The speaker is unsure of the next steps, particularly regarding the integration and the meaning of P(+). They mention the need to consult a source or ask for clarification from an instructor.
  • #1
jb646
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Homework Statement


Starting from Schrodinger's Equation show that a complex potential energy V=α+iβ yields a time dependent probability P of finding the particle in (-inf,inf), i.e. the particle is unstable and normalization cannot be insured over time. Compute P(+)


Homework Equations


SE: ih/2m=-h^2/2m(∫∂ψ/∂x )+Vψ

The Attempt at a Solution


I tried inserting my value of potential energy into the SE as follows
ih/2m=-h^2/2m(∫∂ψ/∂x )+(α+iβ)ψ
and then i got stuck, am i supposed to integrate next or stick an an A and try to normalize and then prove that because it is not normalizeable it is unstable. Also what is P(+)

Thanks for any advice you can offer me.
 
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  • #2
I don't know what they mean by P(+). Look it up in the book / ask your instructor/adviser.

Well, ensuring the conservation of probability (density) over time relies heavily on the fact that this follows from a unitary evolution of states. What you can do is to show that the Hamiltonian will no longer be self-adjoint, since that wouldn't allow us to use Stone's theorem to obtain a unitary evolution.
 

1. What is a complex PE?

A complex PE, or complex potential energy, is a mathematical function used to describe the energy of a system in quantum mechanics. It includes both the real and imaginary components of the potential energy.

2. How does a complex PE affect the time dependence of a particle?

A complex PE introduces an imaginary term in the Schrödinger equation, which results in a time-dependent probability of finding a particle. This means that the probability of finding the particle changes over time.

3. Can you provide an example of a complex PE and its time-dependent probability?

One example is the quantum harmonic oscillator, where the complex PE is given by V(x) = ½ mω²x² + iαx. This results in a time-dependent probability described by the wave function Ψ(x,t) = Ae^(-iωt)e^(-α²t/2)Hn(x-αt), where Hn(x-αt) is the Hermite polynomial.

4. What is the significance of a time-dependent probability in quantum mechanics?

A time-dependent probability allows us to understand the behavior of particles in quantum systems, as it shows how the probability of finding a particle changes over time. This is important in predicting and analyzing the behavior of particles in complex systems.

5. Are there any practical applications of complex PEs and time-dependent probabilities?

Yes, complex PEs and time-dependent probabilities are used in a variety of practical applications, such as in understanding the behavior of electrons in semiconductors and in the development of quantum computing algorithms. They also play a crucial role in studying the dynamics of chemical reactions and in the design of new materials with specific properties.

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