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')##.
  • #1
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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1. What is the wave function for a particle in one dimension in Dirac formalism?

The wave function for a particle in one dimension in Dirac formalism is a mathematical function that describes the quantum state of a particle in one dimension. It is represented by the Greek letter Psi (Ψ) and is a complex-valued function that contains all the information about the position, momentum, and other physical properties of the particle.

2. How is the wave function different in Dirac formalism compared to other formalisms?

The wave function in Dirac formalism is different from other formalisms because it is a spinor, meaning it is a two-component complex-valued function. This allows it to describe particles with intrinsic spin, such as electrons, which cannot be described by a single-component wave function in other formalisms.

3. What is the significance of the Dirac equation in relation to the wave function?

The Dirac equation is a relativistic wave equation that describes the behavior of spin-1/2 particles, such as electrons, in quantum mechanics. It is the fundamental equation in Dirac formalism and provides a way to calculate the wave function for these particles. It also led to the prediction of antimatter and has been crucial in the development of quantum field theory.

4. How does the wave function evolve in time according to Dirac formalism?

In Dirac formalism, the wave function evolves in time according to the Schrödinger equation, which is a partial differential equation that describes how the wave function changes over time. This equation takes into account the particle's energy and potential energy, allowing for the prediction of the particle's behavior at different points in time.

5. Can the wave function for a particle in one dimension in Dirac formalism be measured?

No, the wave function itself cannot be measured. It is a mathematical representation of the quantum state of a particle and does not have a physical interpretation. However, the square of the wave function, known as the probability density, can be measured and provides information about the likelihood of finding the particle at a certain position or momentum.

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