About Schrödinger-Heisenberg connection

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In summary, the conversation discusses the use of matrices and exponential terms in quantum mechanics. The U's are matrices and H is an operator. The exponential terms are defined through series expansion and can be used to define functions of matrices. The use of spectral decomposition is also mentioned as an alternative method for defining functions of operators.
  • #1
snoopies622
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I stumbled over something reading Green's Matrix Mechanics (1965) this afternoon. There was an equation very similar to one I saw in Dirac's Lectures on Quantum Field Theory (1966), where he talks about the equivalence (or near equivalence) of the Schrödinger and Heisenberg formulations of ordinary quantum mechanics:

[tex]

U_{S} = e ^ {-iHt/ \hbar} U_{H} e^ {iHt/ \hbar }

[/tex]

I take it that the U's are matrices, but what are the exponential terms? Vectors? Other matrices? If H is a matrix, what does it mean to raise a real number (e) to the power of a matrix? If instead they are real numbers, wouldn't the two exponential terms then simply cancel each other out?
 
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  • #3
They are operators. H is an operator, the exponential of an operator is defined through the series expansion as an infinite sum. Of course, technicalities such as convergence, domain mustn't be overlooked.
 
  • #4
Hey thanks, I had no idea!
 
  • #5
Usually a function of an (hermitean) operator is defined using the spectral decomposition. Let A be an hermitean operator, let a be its eigenvalues and let |a> define the eigenvectors = an orthonormal basis.

Then

[tex]f(A) = f(A) \sum_a|a\rangle\langle a| = \sum_a f(a)\,|a\rangle\langle a|[/tex]
 
  • #6
Interesting . . Will try this on a few very simple examples and see what happens.
 

FAQ: About Schrödinger-Heisenberg connection

What is the Schrödinger-Heisenberg connection?

The Schrödinger-Heisenberg connection is a mathematical relationship between two fundamental theories of quantum mechanics: the Schrödinger equation, which describes the time evolution of quantum systems, and the Heisenberg uncertainty principle, which states that the position and momentum of a particle cannot be precisely determined simultaneously.

How do the Schrödinger equation and Heisenberg uncertainty principle relate to each other?

The Schrödinger equation and Heisenberg uncertainty principle are complementary concepts that together form the foundation of quantum mechanics. The Schrödinger equation explains how the properties of a quantum system change over time, while the Heisenberg uncertainty principle sets a limit on our ability to know certain properties of a particle at the same time.

Why is the Schrödinger-Heisenberg connection important?

The Schrödinger-Heisenberg connection is important because it helps us understand and predict the behavior of quantum systems. By combining the Schrödinger equation and Heisenberg uncertainty principle, we can make accurate calculations and predictions about the behavior of particles at the atomic and subatomic level.

How did the Schrödinger-Heisenberg connection come about?

The Schrödinger-Heisenberg connection was developed in the early 20th century by Austrian physicist Erwin Schrödinger and German physicist Werner Heisenberg. Both scientists were trying to understand the behavior of atoms and particles at the quantum level, and their work ultimately led to the development of quantum mechanics and the Schrödinger equation and Heisenberg uncertainty principle.

Can you provide an example of the Schrödinger-Heisenberg connection in action?

Yes, one example of the Schrödinger-Heisenberg connection in action is the prediction of the energy levels of an electron in a hydrogen atom. The Schrödinger equation is used to calculate the energy levels, while the Heisenberg uncertainty principle sets a limit on the precision of these calculations. This connection allows scientists to accurately predict and understand the behavior of electrons in atoms, which has many practical applications in fields such as chemistry and material science.

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