Linear superposition of single-particle states

[Neitrino]

Dear all,

I am not sure whether I understand correctly or not.
So from Peskin Schroeder’s book:
$$\phi(x)|0>= \int{\frac{d^3 p}{(2\pi)^3}\frac{1}{2E_p}e^{-ipx}|p>$$
formula (2.41). Interpreting this formula they say – it’s a linear superposition of single-particle states that have well defined momentum. And also that operator phi(x) acting on the vacuum, creates a particle at position x.
My question – since it is a superposition of single-particle states and creates a particle at position X, So that operator creates many single-particle states with different momentum (since there is integration over p and each single-particle state has different momentum) and all of them (particles with different momentum ) are created at one position X?
Or briefly – many different momentum particles are created at one position X?

Thanks

 

[vanesch]

My question – since it is a superposition of single-particle states and creates a particle at position X, So that operator creates many single-particle states with different momentum (since there is integration over p and each single-particle state has different momentum) and all of them (particles with different momentum ) are created at one position X?
Or briefly – many different momentum particles are created at one position X?

What's the superposition of a one-particle and a one-particle state ? A two-particle state or another one-particle state ? Answer: another one-particle state. Superpositions of N-particle states are again N-particle states.
So you should view this as ONE particle is created, in a superposition of momentum states, exactly as in NR quantum mechanics, where ONE position state is written as (about the same) superposition of several momentum states.

cheers,
Patrick.

 

[denian]

for your question. Linear superposition of single-particle states is a fundamental concept in quantum mechanics, and it can be a bit confusing at first. Let me try to provide some clarification.

First of all, the equation you mentioned, \phi(x)|0>= \int{\frac{d^3 p}{(2\pi)^3}\frac{1}{2E_p}e^{-ipx}|p> (formula 2.41), is known as the field operator. This operator acts on the vacuum state |0> to create a particle at position x. However, as you correctly pointed out, this equation represents a superposition of single-particle states, each with a different momentum.

To understand this better, let's break down the equation. The integral \int{\frac{d^3 p}{(2\pi)^3}\frac{1}{2E_p}e^{-ipx}|p> represents a sum over all possible momentum states, with each state weighted by a factor of \frac{1}{2E_p}. This means that the higher the momentum of a state, the smaller its contribution to the overall superposition.

Now, to answer your question, when the field operator acts on the vacuum state, it creates a superposition of all possible single-particle states with different momenta, but all at the same position x. This means that at position x, there is a particle with a certain probability of having a certain momentum. In other words, the particle is in a state of superposition, with different momenta being represented by different terms in the equation.

I hope this helps clarify the concept of linear superposition of single-particle states. It is a fundamental principle in quantum mechanics and plays a crucial role in understanding the behavior of particles at the subatomic level. If you have any further questions, please don't hesitate to ask.