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QUESTION 1: Hydrogen atom in a general state (ignoring spin): The orthonormal energy eigenstates of the hydrogen atom Un,1,m are labelled by the principal quantum number n and the orbital angular momentum quantum numbers I and m. In the following you may write the energy eigenvalues as En ₁. The state of a hydrogen atom at time t = 0 is given by a linear combination of energy eigenstates n² 1 V = (√51,0,0 -√32,1,1 + 1/3,2,-2). 3 (a) Write down the wave function for later times t> 0 assuming the atom is undisturbed. (b) Show that this state is correctly normalised. (c) Find the expectation values of the energy and Îz if the system is in the state V. with respect to 2² Hint: No integrations needed, just use the known eigenvalues of n,1,m and write the energy eigenvalues as En = n² and Î

QUESTION 1: Hydrogen atom in a general state (ignoring spin): The orthonormal energy eigenstates of the hydrogen atom Un,1,m are labelled by the principal quantum number n and the orbital angular momentum quantum numbers I and m. In the following you may write the energy eigenvalues as En ₁. The state of a hydrogen atom at time t = 0 is given by a linear combination of energy eigenstates n² 1 V = (√51,0,0 -√32,1,1 + 1/3,2,-2). 3 (a) Write down the wave function for later times t> 0 assuming the atom is undisturbed. (b) Show that this state is correctly normalised. (c) Find the expectation values of the energy and Îz if the system is in the state V. with respect to 2² Hint: No integrations needed, just use the known eigenvalues of n,1,m and write the energy eigenvalues as En = n² and Î

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QUESTION 1: Hydrogen atom in a general state (ignoring spin): The orthonormal energy eigenstates of the hydrogen atom n,,m are labelled by the principal quantum number n and the orbital angular momentum quantum numbers I and m. In the following you may write the energy eigenvalues as En E. The state of a hydrogen atom at time t = 0 is given by a linear combination of energy eigenstates V (√561,0,0 — √342,1,1 + i√3,2,-2) · (a) Write down the wave function for later times t> 0 assuming the atom is undisturbed. (b) Show that this state is correctly normalised. (c) Find the expectation values of the energy and Îz if the system is in the state V. 2² Hint: No integrations needed, just use the known eigenvalues of Un,1,m with respect to and write the energy eigenvalues as En = ₁. n². = and Îz,
Transcribed Image Text:QUESTION 1: Hydrogen atom in a general state (ignoring spin): The orthonormal energy eigenstates of the hydrogen atom n,,m are labelled by the principal quantum number n and the orbital angular momentum quantum numbers I and m. In the following you may write the energy eigenvalues as En E. The state of a hydrogen atom at time t = 0 is given by a linear combination of energy eigenstates V (√561,0,0 — √342,1,1 + i√3,2,-2) · (a) Write down the wave function for later times t> 0 assuming the atom is undisturbed. (b) Show that this state is correctly normalised. (c) Find the expectation values of the energy and Îz if the system is in the state V. 2² Hint: No integrations needed, just use the known eigenvalues of Un,1,m with respect to and write the energy eigenvalues as En = ₁. n². = and Îz,
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