Understanding <q|p>=exp(ip.x): The Explanation You've Been Searching For

  • Thread starter Colin
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In summary, the conversation discusses the eigenstates of the momentum and position operators in quantum mechanics. The normalization condition for an eigenstate of the momentum operator is <q|p>=2πħδ(q-p), while the scalar product is zero for an eigenstate of the position operator. The correct expression for <x|p> is exp(ipx), and this is derived using P|p>=p|p> and <x|P|p>=-iħd/dx<x|p>. The conversation also mentions the book Quantum Field Theory in a Nutshell by Prof Zee.
  • #1
Colin
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Can anyone explain why?

<q|p>=exp(ip.x)

thanks
 
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  • #2
Depends on the nature of your function to begin with.. more detail needed.. probably due to orthogonality condition
 
  • #3
Colin said:
Can anyone explain why?

<q|p>=exp(ip.x)

thanks


What is [itex] <q| [/itex],an eigenstate of the momentum operator,or of the position operator (actually of their adjoints,but,because they are selfadjoint,you can think that way as well)?
If it's for the momentum operator,then [tex] |p> [/itex] is an eigenstate as well,and the normalization condition reads:
[tex] <q|p> =2\pi\hbar \delta(q-p) [/tex]
,but if it's for the coordinate operator,then the scalar product is zero,since one of them is an eigenvector from a Hilbert space and the other is a linear functional over another Hilbert space.

So,anyway,what u've written there is wrong.

Daniel.
 
  • #4
correction <x|p>=exp(ip.x)

sorry I meant <x|p>=exp(ip.x)

and I think I have it

P|p>=p|p>

<x|P|p> =p<x|p>=-id/dx<x|p>
solution
<x|p>=exp(ipx)

It's a long time since I did QM, and I started reading Prof Zee's Quantum field theory in a nut shell. He quotes this very early on without proof and I couldn't see where it came from. Thanks for comments.
 

FAQ: Understanding <q|p>=exp(ip.x): The Explanation You've Been Searching For

1. What is the significance of the equation =exp(ip.x)?

The equation =exp(ip.x) is a fundamental equation in quantum mechanics that describes the relationship between the position and momentum of a particle. It is known as the Heisenberg uncertainty principle and states that the more accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa.

2. How does this equation relate to the wave-particle duality of quantum mechanics?

The equation =exp(ip.x) is derived from the wave-particle duality of quantum mechanics. It shows that particles can exhibit both wave-like and particle-like behavior, and that the position and momentum of a particle are described by a wave function. This equation is essential in understanding the behavior of particles at the quantum level.

3. Can you explain the mathematical meaning of the equation =exp(ip.x)?

The equation =exp(ip.x) is a mathematical representation of the uncertainty principle. The symbol represents the expectation value or average value of the position and momentum of a particle, while exp(ip.x) is the mathematical expression for a wave function. The equation shows that the product of the uncertainty in position and momentum is equal to Planck's constant divided by 2*pi.

4. What practical applications does this equation have?

The equation =exp(ip.x) has many practical applications in fields such as quantum computing, cryptography, and particle physics. It is also used in the development of technologies such as MRI machines and atomic clocks. Understanding this equation is crucial for advancements in these fields and for the development of new technologies.

5. How can I apply this equation in my own research or studies?

If you are studying or conducting research in the fields of quantum mechanics, particle physics, or other related fields, understanding the equation =exp(ip.x) is essential. It can help you make predictions and calculations about the behavior of particles and their properties. Additionally, knowing this equation can provide a deeper understanding of the fundamental principles of quantum mechanics.

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