Solving Bra-ket Multiplication: <n|(a-+a+)^4|n> = 5n2+5n+3

[crystalplane]

hey guys,

<n|(a^-+a⁺)^4|n> = 5n²+5n+3

I am wondering if some one can show me why left and right side are equal

　　　　Thanks in advance

 

[Pengwuino]

If it is true, it's simply a matter of doing the work.

$$(a^ - + a^ + )^2 = (a^ - a^ - + a^ - a^ + + a^ + a^ - + a^ + a^ + )$$ for example, so you do the case for the 4th power, you can work out what the result is knowing
$$\[ \begin{array}{l} a^ + \left. {|n} \right\rangle = \sqrt {n + 1} \left. {|n + 1} \right\rangle \\ a^ - \left. {|n} \right\rangle = \sqrt n \left. {|n - 1} \right\rangle \\ \end{array}$$

and any term that doesn't return the ket to |n> can be ignored since it'll go away.

 

[Entropia]

!

Hello,

I would be happy to explain the solution to this equation. First, let's break down the components of the equation. The <n| represents the bra vector, which is the complex conjugate of the ket vector |n>. The (a-+a+)^4 represents the operator, which in this case is a combination of the annihilation operator (a-) and creation operator (a+), raised to the fourth power. Finally, the |n> represents the ket vector.

To solve this equation, we can use the properties of the annihilation and creation operators. First, we know that the annihilation and creation operators are Hermitian conjugates of each other, meaning that (a-)^† = a+. This allows us to simplify the equation to <n|(a+)^4|n>.

Next, we can use the commutation relation between the annihilation and creation operators, [a-, a+] = 1, to expand the operator (a+)^4. This gives us <n|(a+)^4|n> = <n|(a+)(a+)(a+)(a+)|n> = <n|(a+)(a+)(a+)(a+)|n> = <n|(a+)(a+)^3|n>.

Using the commutation relation again, we can further simplify the equation to <n|(a+)^3(a+)|n> = <n|(a+)^2(a+)^2|n> = <n|(a+)^4|n>.

Finally, we can use the fact that the annihilation and creation operators commute with the bra and ket vectors to simplify the equation to <n|(a+)^4|n> = (a+)4<n|n> = 4<n|n>.

Since <n|n> is equal to 1, we can conclude that <n|(a-+a+)^4|n> = 4. However, the equation provided states that the result is 5n2+5n+3, which is not equal to 4. Therefore, there must be a mistake in the equation or the solution.

I hope this explanation helps. Please let me know if you have any further questions.