Raising and lowering operators

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In summary, the conversation discusses the expectation value of a product of 4 raising and lowering operators with a lowering operator on the extreme right in the ground state of a harmonic oscillator. It is explained that when the lowering operator operates on the first excited state, it results in the ground state. When it operates on the ground state, it results in zero. This means that the product cannot be calculated as the lowering operator has nothing to operate on.
  • #1
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Homework Statement



Can anyone please explain why any term which is a product of 4 raising and lowering operators with a lowering operator on the extreme right (eg. A-A+A+A-) has zero expectation value in the ground state of a harmonic oscillator?



Homework Equations





The Attempt at a Solution

 
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  • #2
What happens if you let the lowering operator work on [itex]\psi_1[/itex]? What do you get when you let the lowering operator work on [itex]\psi_0[/itex]?

Calculate [itex]a \psi_1[/itex] and [itex]a \psi_0[/itex]. Does [itex]a \psi_0[/itex] exist?
 
  • #3
I get it now... thank you. When the lowering operator operates on v1, I get v0. When it operates on v0, I get zero. A- v0 doesn't exist so the rest of the operators in the product have nothing to operate on.

Cheers.
 

1. What are Raising and Lowering Operators?

Raising and lowering operators are mathematical tools used in quantum mechanics to describe the behavior of subatomic particles. They are used to describe the creation and annihilation of particles, and to calculate the energy states of a quantum system.

2. How do Raising and Lowering Operators work?

Raising operators increase the energy state of a system by one unit, while lowering operators decrease the energy state by one unit. They are represented by mathematical operators that act on the wave function of a particle. By applying these operators, we can calculate the possible energy states of a quantum system.

3. What is the difference between Raising and Lowering Operators?

The main difference between raising and lowering operators is the direction in which they act on the wave function. Raising operators increase the energy state, while lowering operators decrease it. They are also represented by different mathematical symbols, with raising operators often denoted by the symbol "+", and lowering operators denoted by the symbol "-".

4. What are some examples of Raising and Lowering Operators?

One example of a raising operator is the creation operator, which is used to describe the creation of a particle in a higher energy state. An example of a lowering operator is the annihilation operator, which describes the destruction of a particle in a lower energy state. Other examples include the ladder operators used in the harmonic oscillator model and the spin operators used to describe the spin of particles.

5. What is the importance of Raising and Lowering Operators in quantum mechanics?

Raising and lowering operators are essential tools in quantum mechanics, as they allow us to describe and calculate the energy states of a quantum system. They also play a crucial role in the study of quantum field theory, which is used to describe the behavior of particles at the subatomic level. Without raising and lowering operators, it would be challenging to understand and predict the behavior of subatomic particles.

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