How Is the Wavefunction Ψ(x) Derived from Momentum Space?

Therefore, the integral becomes ∫dp exp{ipx/h}Ψ(p).As for your second question, Ψ(x) is not necessarily equal to <x|\psi>, but it is common to use the same symbol for the wave function and the state in the position basis. It is just a matter of notation and convenience.
  • #1
Eole
2
0
Here's a derivation of wavefunction of State Ψ in representations of coordinates and momentum
Ψ (x)=<x|Ψ >=<x|∫dp|p><p|Ψ >=∫dp<x|p><p|Ψ>=∫dp exp{ipx/h}Ψ(p)
Ψ (p)=<p|Ψ >=∫dx exp{-ipx/h}Ψ(x)


Ψ (x)=<x|Ψ >=<x|∫dp|p><p|Ψ >=∫dp<x|p><p|Ψ>=∫dp exp{ipx/h}Ψ(p)
i don't understand how ∫dp<x|p><p|Ψ> become ∫dp exp{ipx/h}Ψ(p)
Could you please tell me the drivation of this formula?

and another question is why Ψ (x) could be denoted as <x|Ψ >?
 
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  • #2
I probably won't be able to help, but I'm sure that if I'm struggling to make sense of that, then other people are too. Some things just can't be written without proper equations. Maybe try writing the equation on Microsoft word's equation tool and attach the document.
 
  • #3
Eole said:
Here's a derivation of wavefunction of State Ψ in representations of coordinates and momentum
Ψ (x)=<x|Ψ >=<x|∫dp|p><p|Ψ >=∫dp<x|p><p|Ψ>=∫dp exp{ipx/h}Ψ(p)
Ψ (p)=<p|Ψ >=∫dx exp{-ipx/h}Ψ(x)
Ψ (x)=<x|Ψ >=<x|∫dp|p><p|Ψ >=∫dp<x|p><p|Ψ>=∫dp exp{ipx/h}Ψ(p)
i don't understand how ∫dp<x|p><p|Ψ> become ∫dp exp{ipx/h}Ψ(p)
Could you please tell me the drivation of this formula?
and another question is why Ψ (x) could be denoted as <x|Ψ >?

It seems to me that you are asking why
<x|p>=exp{ipx/h}
but surely you must have this derivation in your notes? BTW, the above equation isn't actually quite right (needs a factor in order to normalize it) and, in my notes anyway, the 'h' is actually an 'h-bar'.

As regards your other question, I can only assume that it is very sloppy notation because I have never seen a state use the same symbol as the wave function (although I am still an undergraduate so my experience is rather limited).
 
  • #4
Eole said:
Ψ (x)=<x|Ψ >=<x|∫dp|p><p|Ψ >=∫dp<x|p><p|Ψ>=∫dp exp{ipx/h}Ψ(p)
i don't understand how ∫dp<x|p><p|Ψ> become ∫dp exp{ipx/h}Ψ(p)
Could you please tell me the drivation of this formula?
and another question is why Ψ (x) could be denoted as <x|Ψ >?

[itex]\psi(x) = \langle x | \psi \rangle[/itex] by definition. It is the component of |x> in the expansion of [itex]| \psi \rangle[/itex] in the position basis.

∫dp<x|p><p|Ψ> = ∫dp exp{ipx/h}Ψ(p), because [itex]\langle x|p \rangle =\exp{ipx/\hbar}[/itex] (apart from a constant factor) and <p|Ψ>=Ψ(p), again by definition.
 

FAQ: How Is the Wavefunction Ψ(x) Derived from Momentum Space?

1. What is a wavefunction?

A wavefunction is a mathematical function that describes the quantum state of a particle or system. It contains information about the position, momentum, and other physical properties of the particle.

2. How is a wavefunction different from a regular function?

A wavefunction is different from a regular function because it is used to describe the behavior of quantum particles, which exhibit wave-like properties. Unlike regular functions, wavefunctions can have complex values and are subject to the rules of quantum mechanics.

3. What is the importance of the wavefunction in quantum mechanics?

The wavefunction is essential in quantum mechanics as it is used to calculate the probability of finding a particle in a certain position or state. It also allows us to describe the behavior of particles at the quantum level and understand phenomena such as superposition and entanglement.

4. How is the wavefunction related to Schrödinger's equation?

Schrödinger's equation is a mathematical equation that describes the time evolution of a quantum system. The wavefunction of the system is a solution to this equation, and it can be used to predict the future behavior of the system.

5. Can the wavefunction be observed or measured?

No, the wavefunction itself cannot be observed or measured. It is a mathematical representation of a particle's quantum state and exists in an abstract mathematical space. However, we can measure the effects of the wavefunction, such as the probability of finding a particle in a certain location.

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