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ƯU = I = UUt where I is the identity operator. (a) Show that if (v, 6) = 1 then (Up,Uv) = 1, i.e the operation of U on v preserves the norm of the wavefunction. (b) Using the general properties established in question 6.2 above for adjoint operators, show for any operator A that C = i(A – A†) is a hermitian operator, i.e. C = C†. (c) Show that if H is a hermitian operator, then the adjoint operator of expiH, where expiH is defined to be exp(iH) = H", is the operator exp(-iH). (Note that expiH is a unitary %3D n! n=0 operator.

ƯU = I = UUt where I is the identity operator. (a) Show that if (v, 6) = 1 then (Up,Uv) = 1, i.e the operation of U on v preserves the norm of the wavefunction. (b) Using the general properties established in question 6.2 above for adjoint operators, show for any operator A that C = i(A – A†) is a hermitian operator, i.e. C = C†. (c) Show that if H is a hermitian operator, then the adjoint operator of expiH, where expiH is defined to be exp(iH) = H", is the operator exp(-iH). (Note that expiH is a unitary %3D n! n=0 operator.

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Quantum Theory

Second picture is the properties for part b)

(A+ B)* = AÏ + B*, + B', (AB)† = B† A†, (AA)† = XA',
Transcribed Image Text:(A+ B)* = AÏ + B*, + B', (AB)† = B† A†, (AA)† = XA',
Q6:3 Another important type of operator in quantum theory is a so-called unitary operator, U, which has the property U*U = I = UU* where I is the identity operator. (a) Show that if (b, v) = 1 then (U, U%) = 1, i.e the operation of U on v preserves the norm of the wavefunction. (b) Using the general properties established in question 6.2 above for adjoint operators, show for any operator A that C = i(A – Ai) is a hermitian operator, i.e. C = Ct. (c) Show that if H is a hermitian operator, then the adjoint operator of expiH, where expiH is defined to be exp(iH) = Ï H, is the operator exp(-iH). (Note that expiH is a unitary n=0 operator.)
Transcribed Image Text:Q6:3 Another important type of operator in quantum theory is a so-called unitary operator, U, which has the property U*U = I = UU* where I is the identity operator. (a) Show that if (b, v) = 1 then (U, U%) = 1, i.e the operation of U on v preserves the norm of the wavefunction. (b) Using the general properties established in question 6.2 above for adjoint operators, show for any operator A that C = i(A – Ai) is a hermitian operator, i.e. C = Ct. (c) Show that if H is a hermitian operator, then the adjoint operator of expiH, where expiH is defined to be exp(iH) = Ï H, is the operator exp(-iH). (Note that expiH is a unitary n=0 operator.)
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