Second Quantization: Motivation & Field Operators

In summary, Second Quantization is a mathematical trick used to write many-particle wave functions in a symmetric or antisymmetric form, making it useful for describing the creation and annihilation of particles. It is essential in understanding quantum statistical mechanics and simplifies the formulation using fields. The terminology may be controversial, but it is widely used in the field.
  • #1
Euphemia
12
0
Hi,all
What is the motivation of using Second Quantization ?
What kind of situation does people want to use field operators ?
Euphemia
 
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  • #2
In non-relativistic quantum mechanics, second quantization is mathematically equivalent to the Schroedinger equation. Take a lattice of atoms and write the Schroedinger equation. Second quantization essentially just taking the Fourier transform, and the "particles" are Fourier modes. Since it is natural for Fourier modes to be created and destroyed, these are useful for describing "particles" which are created and destroyed. In the relativistic case, one postulates the second quantized version directly without an underlying lattice.
 
  • #3
The framework of non-rel. QM based on the Schroedinger equation is not suitabe for the description of creation and annihilation of particles, e.g. electron - positron annihilation (and production of two photons). In order to achieve that one needs 'second quantization'
 
  • #4
In non-relativistic QM, second quantization is a mathematical trick useful to write many-particle wave functions in a symmetric (Bose-Einstein statistics) or antisymmetric (Fermi-Dirac statistic) form automatically.
 
  • #5
There's no <second> quantization, there's only <quantization> which means passing from classical Hamiltonian observables to quantum counterparts where they exist and are uniquely defined (Groeneveld-van Hove problem).
 
  • #7
Calculating Occupation Numbers. It's a math trick at best.
 
  • #8
dextercioby said:
There's no <second> quantization, there's only <quantization> which means passing from classical Hamiltonian observables to quantum counterparts where they exist and are uniquely defined (Groeneveld-van Hove problem).

Meaning you can either see it as a one time thing (and then face the question, why stop there?), or (the way I prefer it) as defining an induction step and then worry about howto interpret that tower of turtles.

http://math.ucr.edu/home/baez/nth_quantization.html contains some interesting ponderings.

/Fredrik
 
  • #9
Euphemia said:
What is the motivation of using Second Quantization ?
What kind of situation does people want to use field operators ?

It is very basic for an understanding of quantum statistical mechanics, where almost everything gets much simpler in the second quantization formulation using fields.

Some people don't like the terminology, but it is very standard.
(But one should not imagine that it is literally a quantization of something already quantized.)
 

What is second quantization?

Second quantization is a mathematical framework used in quantum mechanics to describe systems with many particles. It is based on the concept of creating and destroying particles, and is often used to study systems with a large number of particles, such as solids and fluids.

What is the motivation behind second quantization?

The motivation behind second quantization is to simplify the mathematical description of systems with many particles. By introducing creation and annihilation operators, the equations can be written in a more compact form, making it easier to handle large systems with complex interactions.

What are field operators in second quantization?

Field operators are operators that describe the behavior of particles in a quantum field theory. In second quantization, field operators are used to create and destroy particles in a given quantum state. They are expressed in terms of creation and annihilation operators.

How is second quantization different from first quantization?

First quantization describes the behavior of individual particles, while second quantization describes the behavior of many particles in a single mathematical framework. First quantization uses wave functions to describe particles, while second quantization uses field operators.

What are some applications of second quantization?

Second quantization is used in many areas of physics, such as solid state physics, nuclear physics, and quantum field theory. It is especially useful for studying systems with a large number of interacting particles, such as superconductors and superfluids.

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