A three dimenisonale quantum systeme,

vector basis state is |0.), |e.), |2e.). NOTE: A is A-HAT

Values for A-hat_1 = 2 | 0 > , A-hat_2 = 2i | e >, A-hat_3 = 3 | 2 e >

Basis states

A-hat_1 | 0 > = 2 | 0 > - 3 | e >

A-hat_2 | e > = - 2 | 0 > + | e > - 2i | 2e >

A-hat_3 | 2e > = 2i | e > + 4 | 2e >

i) Find state vector | >'when A-hat_1 = 2, A-hat_2 = 2i and A-hat.3 = 3 and also the normalized form | >. Please carefully explain formula or instructions.

ii) Using the normalized form found in part i, evaulate | > = A-hat | > using dirac notation. and then calcualte

< | A-hat |>.Please carefully explain all steps

iii) Discover matrix represntation A for A-hat operator. Please explain all steps

iv) Discover outer product for A-hat. Please explain all steps

Appreciate if you can attempt all questions. Thank you. If you have any questions before attempting, please ask and I will provide more information to you