What is the Wave Function for a Particle in One Dimension in Dirac Formalism?

In summary, the expression ##<x|P|x'>## represents the momentum operator in one dimension when the value of Planck's constant is set to 1. It can be evaluated using integration by parts, resulting in the expression ##-i\frac{\partial}{\partial x}\delta(x-x')##.
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ergospherical
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What is ##<x|P|x'>##? (for particle in 1d, and ##\hbar = 1##)?\begin{align*}
<x|P|x'> &= \int dp' <x|P|p'><p'|x'> \\
&= \int dp' \ p' <x|p'> <p'|x'> \\
&= \int dp' \ p' \frac{1}{\sqrt{2\pi}} e^{ip'x} \frac{1}{\sqrt{2\pi}} e^{-ip'x'} \\
&= \frac{1}{2\pi} \int dp' \ p' e^{ip'(x-x')}
\end{align*}
 
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  • #2
ergospherical said:
What is ##<x|P|x'>##? (for particle in 1d, and ##\hbar = 1##)?\begin{align*}
<x|P|x'> &= \int dp' <x|P|p'><p'|x'> \\
&= \int dp' \ p' <x|p'> <p'|x'> \\
&= \int dp' \ p' \frac{1}{\sqrt{2\pi}} e^{ip'x} \frac{1}{\sqrt{2\pi}} e^{-ip'x'} \\
&= \frac{1}{2\pi} \int dp' \ p' e^{ip'(x-x')}
\end{align*}
I'm not sure what the question is? So far so good. Now integrate by parts.

-Dan
 
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  • #3
it's supposed to evaluate to \begin{align*}
-i \frac{\partial}{\partial x} \delta(x-x')
\end{align*}but even integrating by parts I'm not sure how to get this
 
  • #4
ergospherical said:
What is ##<x|P|x'>##? (for particle in 1d, and ##\hbar = 1##)?\begin{align*}
<x|P|x'> &= \int dp' <x|P|p'><p'|x'> \\
&= \int dp' \ p' <x|p'> <p'|x'> \\
&= \int dp' \ p' \frac{1}{\sqrt{2\pi}} e^{ip'x} \frac{1}{\sqrt{2\pi}} e^{-ip'x'} \\
&= \frac{1}{2\pi} \int dp' \ p' e^{ip'(x-x')}
\end{align*}
From this you get
$$\langle x|\hat{P}|x' \rangle=\frac{1}{2 \pi} (-\mathrm{i} \partial_x) \int_{\mathbb{R}} \mathrm{d} p' \exp[\mathrm{i} p' (x-x')]=-\mathrm{i} \partial_x \delta(x-x').$$
 
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FAQ: What is the Wave Function for a Particle in One Dimension in Dirac Formalism?

What is the wave function in Dirac formalism?

In Dirac formalism, the wave function of a particle is represented by a state vector, denoted as |ψ⟩, in a complex Hilbert space. This state vector contains all the information about the quantum state of the particle.

How is the wave function in one dimension expressed in Dirac notation?

In one dimension, the wave function ψ(x) can be expressed in Dirac notation as ⟨x|ψ⟩, where |x⟩ represents the position eigenstate. The function ψ(x) is the projection of the state vector |ψ⟩ onto the position basis |x⟩.

What is the significance of the position basis in Dirac formalism?

The position basis |x⟩ is crucial in Dirac formalism because it allows us to describe the wave function in terms of position coordinates. The state vector |ψ⟩ can be expanded in terms of the position basis as |ψ⟩ = ∫ψ(x)|x⟩ dx, where ψ(x) = ⟨x|ψ⟩.

How do you normalize the wave function in Dirac formalism?

To normalize the wave function in Dirac formalism, the integral of the absolute square of the wave function over all space must equal one: ∫|ψ(x)|² dx = 1. This ensures that the total probability of finding the particle somewhere in space is one.

What is the role of the Hamiltonian operator in Dirac formalism?

The Hamiltonian operator Ĥ plays a central role in Dirac formalism as it governs the time evolution of the wave function. The time-dependent Schrödinger equation is Ĥ|ψ(t)⟩ = iħ (d/dt)|ψ(t)⟩, which describes how the state vector |ψ(t)⟩ evolves over time.

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