in quantum mechanics 1- calculate the matrix (m|p2|n > and (m|x2|n > then find the matrix of kinetic and potentialenergy of SHM

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in quantum mechanics

1- calculate the matrix (m|p2|n > and (m|x2|n > then find the matrix of kinetic and potential
energy of SHM

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Step 1

The position and momentum operator in terms of ladder operator are defined as :

$x \to \hat{x} = \sqrt{\frac{ħ}{2 m ω}} (a + a^{†}) p \to \hat{p} = - i \sqrt{\frac{ħ m ω}{2}} (a - a^{†}) x^{2} = \frac{ħ}{2 m ω} (a + a^{†}) (a + a^{†}) = \frac{ħ}{2 m ω} (a a + a a^{†} + a^{†} a + a^{†} a^{†}) p^{2} = - \frac{ħ m ω}{2} (a - a^{†}) (a - a^{†}) = - \frac{ħ m ω}{2} (a a - a a^{†} - a^{†} a + a^{†} a^{†})$

The definitions of ladder operators are :

$a^{†} | n > = \sqrt{n + 1} | (n + 1) > a | n > = \sqrt{n} | (n - 1) >$

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1- calculate the matrix (m|p²|n > and (m|x²|n > then find the matrix of kinetic and potential energy of SHM