Calculating <E> & <E^2> with Eigenfunctions of Parity Operator

[cuegirl60]

Q1
energy no. of times measured
a1 n1
a2 n2
a3 n3
a4 n4

expectation value <E> = (a1n1+a2n2+a3n3+a4n4) / (n1+n2+n3+n4)
is this correct?

Also, how do you caluculate expectation value <E^2> ?
i.e. <E squared>

Q2
Identify if the following functions are eigenfunctions of the parity operator.

a) f(z) = z(a-z)(z+b), where a,b are real numbers
b) f(x) = Ψ(x)xΨ(x), where Ψ(x) is antisymmetric about the origin.
c) same f(x) in b), but where Ψ(x) is symmetric about the origin.
d) f(x) = Ψ(x)x^2Ψ(x) where Ψ(x) is antisymmetric about the origin. x^2 means x squared.

 

[dextercioby]

Q1
energy no. of times measured
a1 n1
a2 n2
a3 n3
a4 n4

expectation value <E> = (a1n1+a2n2+a3n3+a4n4) / (n1+n2+n3+n4)
is this correct?

Also, how do you caluculate expectation value <E^2> ?
i.e. <E squared>

How did you come up with the first answer ? Follow the same line of thought and you'll find the answer to <E^2>.

Q2
Identify if the following functions are eigenfunctions of the parity operator.

a) f(z) = z(a-z)(z+b), where a,b are real numbers
b) f(x) = Ψ(x)xΨ(x), where Ψ(x) is antisymmetric about the origin.
c) same f(x) in b), but where Ψ(x) is symmetric about the origin.
d) f(x) = Ψ(x)x^2Ψ(x) where Ψ(x) is antisymmetric about the origin. x^2 means x squared.

Post ideas and work, people here want to help, not supply full solutions to your problems.

Daniel.

 

[cuegirl60]

so i said <E> = (a1n1+a2n2+a3n3+a4n4) / (n1+n2+n3+n4)

and if this was correct,

<E^2> = (a1^2 n1 + a2^2 n2 + a3^2 n3 + a4^2 n4) / (n1+n2+n3+n4)

is this correct??

Q2, my thought was:
only d was eigenfunctions of the parity operator.

b,c always give function to the power of odd value, a i just worked out can not be...

 

[OlderDan]

so i said <E> = (a1n1+a2n2+a3n3+a4n4) / (n1+n2+n3+n4)

and if this was correct,

<E^2> = (a1^2 n1 + a2^2 n2 + a3^2 n3 + a4^2 n4) / (n1+n2+n3+n4)

is this correct??

Q2, my thought was:
only d was eigenfunctions of the parity operator.

b,c always give function to the power of odd value, a i just worked out can not be...

Both of these look good.