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Ratzinger
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Would someone mind opening his or her Shankar page 153 and tell me what (5.1.10) and (5.1.11) exactly mean? Does (5.1.10) stand for a delta function?
thank you
thank you
Galileo said:Are these the equations?
(5.1.10)
[tex]U(x,t;x')\equiv \langle x|U(t)|x' \rangle = \int_{-\infty}^{\infty}\langle x|p\rangle \langle p|x' \rangle e^{-ip^2t/2m\hbar}dp
=\frac{1}{2\pi \hbar}\int_{-\infty}^\infty e^{ip(x-x')/\hbar} \cdot e^{-ip^2t/2m\hbar}dp=\left(\frac{m}{2\pi \hbar it}\right)^{1/2}e^{im(x-x')^2/2\hbar t[/tex]
(5.1.11)
[tex]\psi(x,t)=\int U(x,t;x')\psi(x',0)dx'[/tex]
The title refers to a specific section in a book called "Shankar" on page 153, which discusses the meaning of the delta function in mathematics.
The delta function, denoted as δ(x), is a mathematical function that is defined as 0 for all values of x except when x is 0, where it is defined as infinity. Essentially, it is a function that is used to represent an "impulse" or instantaneous change in a system.
The delta function is significant because it allows us to model and analyze systems that involve sudden changes or impulses. It is also used in many areas of physics, engineering, and mathematics, such as signal processing, quantum mechanics, and differential equations.
In mathematics, the delta function is often used as a "distribution" or a generalized function. It is used to define other functions, such as the Heaviside step function and the Dirac delta distribution. It is also used in integration and solving differential equations.
The delta function has many real-world applications, such as in signal processing for analyzing sudden changes in signals, in quantum mechanics for representing "wave packets" of particles, and in engineering for modeling impulse responses in systems. It is also used in probability and statistics for representing discrete probability distributions.