Questions about some material from Dirac

In summary, the standard ket is a vector in Hilbert space that describes the quantum state of a system, with its norm being unity. Schrodinger's representation allows for the use of only position and position derivatives. The coefficient i in the quantum Poisson bracket is the complex i, and arbitrary phase factors cannot be eliminated when they are not constant.
  • #1
JasonJo
429
2
Hey I was hoping you guys could clarify some stuff in Dirac, I'm trying to sort through the Schrodinger representation:

1) What exactly is the standard ket > ? Can anyone give me it in terms of pure linear algebra? What does it mean for it to be unity in terms of wave functions??

2) I guess Schrodinger's representation allows us to use only position and corresponding position derivatives to describe a physical system?

3) When we have the coefficient i in the equation for the quantum Poisson bracket, is this the complex i? Or is it an arbitrary coefficient.

4) When exactly can't we eliminate arbitrary phase factors? I know when the phase factor is a constant, we can take it such that it is unity when it is multiplied by it's conjugate.
 
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  • #2
1) The standard ket is a vector in Hilbert space that describes the quantum state of a system. In terms of linear algebra, it is a vector in an inner product space, with the inner product being the probability amplitude of the system. It is unity if its norm (the absolute value of its inner product with itself) is equal to 1. This means that the amplitude of the wavefunction is equal to 1 for all points in space.2) Yes, Schrodinger's representation allows us to use only position and corresponding position derivatives to describe a physical system.3) Yes, the coefficient i in the equation for the quantum Poisson bracket is the complex i.4) We can't eliminate arbitrary phase factors when the phase factor is not a constant. In this case, we can only take it such that it is equal to 1 when it is multiplied by its conjugate up to a multiplicative constant.
 
  • #3


1) The standard ket, also known as the Dirac notation, is a notation used in quantum mechanics to represent states of a system. In pure linear algebra terms, it can be thought of as a vector in a vector space. The notation is <ψ|, where ψ represents the state. The ket is unity when it is multiplied by its bra, which is the conjugate transpose of the ket, resulting in a scalar value of 1. This represents the normalization of the state in terms of wave functions.

2) Yes, the Schrodinger representation allows us to describe a physical system using only position and corresponding position derivatives. This is because the wave function in this representation is a function of position, making it easier to understand and analyze.

3) The coefficient i in the quantum Poisson bracket is the complex i, also known as the imaginary unit. It is not an arbitrary coefficient, but a fundamental constant in mathematics and physics.

4) We can eliminate arbitrary phase factors when they are constant, meaning they do not depend on the variables of the system. In this case, we can choose a phase factor such that it becomes unity when multiplied by its conjugate. However, if the phase factor is not constant, it cannot be eliminated and must be taken into account in calculations.
 

1. What is the significance of Dirac's equation in quantum mechanics?

Dirac's equation, also known as the Dirac equation of the electron, is a fundamental equation in quantum mechanics that describes the behavior of an electron in a relativistic context. It combines the principles of quantum mechanics and special relativity, providing a more accurate description of the behavior of electrons at high speeds.

2. What is the contribution of Dirac's equation to the understanding of antimatter?

Dirac's equation predicted the existence of antimatter, which was later confirmed by experiments. This equation showed that for every particle, there is a corresponding antiparticle with the same mass and opposite charge. This discovery helped to advance our understanding of the fundamental building blocks of the universe.

3. How does Dirac's equation differ from Schrödinger's equation?

Dirac's equation is a more comprehensive and accurate version of Schrödinger's equation. It includes the principles of special relativity, while Schrödinger's equation only considers non-relativistic systems. Dirac's equation also predicts the existence of antimatter, while Schrödinger's equation does not.

4. What is the role of spin in Dirac's equation?

Dirac's equation incorporates the concept of spin, which is a fundamental property of particles. Spin is a quantum mechanical property that describes the intrinsic angular momentum of a particle. Dirac's equation showed that spin is a necessary component for the description of particles, and it helped to explain some of the anomalies observed in the behavior of electrons.

5. How does Dirac's equation contribute to the development of quantum field theory?

Dirac's equation was a key development in the field of quantum field theory. It provided a mathematical framework for understanding the behavior of particles at high energies and speeds, and it paved the way for the development of other important theories such as quantum electrodynamics and quantum chromodynamics. It also helped to bridge the gap between quantum mechanics and special relativity, leading to a more comprehensive understanding of the fundamental laws of physics.

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