= A. 1) Suppose annihilation (â-) and creation (â+) operator are defined as â+ Calculate the commutator: [â, â+] √zm (Î

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A 1 Suppose Annihilation A And Creation A Operator Are Defined As A Calculate The Commutator A A Zm I 1 (113.28 KiB) Viewed 93 times

= A. 1) Suppose annihilation (â-) and creation (â+) operator are defined as â+ Calculate the commutator: [â, â+] √zm (Î ±imwî). 2) Use â to express the Hamiltonian of a one-dimensional (1D) harmonic oscillator of mass m and natural frequency w. Hence, show that â+ increases the energy of n) by hw amount where n) is the n-th eigenstate of harmonic oscillator corresponding to the energy eigenvalue En = (n + ¹)ħw. 3) Derive the position space representation of ground state wave function of 1D harmonic oscillator by using the fact that â annihilates the ground state of harmonic oscillator.. 4) A one-dimensional harmonic oscillator of mass m and natural frequency w in the quantum state (t = 0)) = 10)+¹) +12) i √3 at time t 0, where n) denotes the eigenstate with eigenvalue (n+1)ħw. Calculate the expectation value (y(t) (t)) of the position operator, at some arbitrary time t. -