Why is <n|n-2> equal to 0 in computing <n|\hat{a}^2|n>?

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In summary, to solve for <n|n-2>, one would set the equation equal to 0 and simplify it by combining like terms. Then, algebraic techniques such as factoring or the quadratic formula can be used to find the value of n. The notation <n|n-2> in this equation represents the dot product of the vectors n and n-2. Setting <n|n-2> equal to 0 is necessary in order to find solutions or equilibrium points. A real-life application of this equation can be seen in quantum mechanics, where it is used to determine the energy levels of a system. While <n|n-2> = 0 may not be the only solution to this equation,
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Kidiz
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Homework Statement



This is not really homework. I'm just studying and came across this question. So, I'm not sure if I should post this question here on one of the physics sections.

On computing ##<n|\hat{a}^2|n>##, one arrives at ##\sqrt{n}*\sqrt{n-1}<n|n-2>=0## because ##<n|n-2>=0##

Now, why is that so?
 
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  • #2
Isn't that just because the two Fock states are orthogonal?
 

FAQ: Why is <n|n-2> equal to 0 in computing <n|\hat{a}^2|n>?

How do you solve for ?

To solve for , you would first set the equation equal to 0. Then, you would simplify the equation by combining like terms. Finally, you would use algebraic techniques such as factoring or the quadratic formula to solve for the value of n.

What does the notation mean in this equation?

The notation represents the inner product or dot product of the vectors n and n-2. This is a mathematical operation that combines two vectors to produce a scalar value.

Why does need to equal 0?

In many mathematical and scientific problems, equations are set equal to 0 in order to find solutions or equilibrium points. In this case, setting equal to 0 allows us to solve for the value of n that satisfies the equation.

Can you give an example of a real-life application of this equation?

One possible application of = 0 is in physics, specifically in the study of quantum mechanics. The equation is used to determine the energy levels of a quantum system, where n represents the principal quantum number and n-2 represents the angular momentum quantum number.

Is equal to 0 the only solution to this equation?

No, there may be other solutions to this equation depending on the specific values of n and n-2. However, setting equal to 0 is often a useful starting point for solving the equation and finding other solutions.

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