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Determining whether an operator is Hermitian

  • Thread starter GoKush
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Homework Statement



Consider the set of functions [itex] {f(x)} [/itex] of the real variable [itex] x [/itex] de fined on the interval [itex] -\infty< x < \infty [/itex] that go
to zero faster than [itex] 1/x [/itex] for [itex]x\rightarrow ±\infty[/itex] , i.e.,
[tex]
\lim_{n\rightarrow ±\infty} {xf(x)}=0
[/tex]
For unit weight function, determine which of the following linear operators is Hermitian when acting upon [itex] {f(x)} [/itex]:
[tex] (a) \frac{d}{dx} + x[/tex] [tex](b) -i \frac{d}{dx}+x^2 [/tex][tex](c) ix \frac{d}{dx} [/tex][tex](d) ix \frac{d^3}{dx^3} .[/tex]


Homework Equations



[itex] Hf(x)=λf(x) [/itex] has real values of [itex] λ [/itex] where [itex] H [/itex] is a Hermitian operator and [itex] λ [/itex] are it's eigenvalues



The Attempt at a Solution



[tex] a) \frac{df(x)}{dx} + xf(x) = λf(x) [/tex][tex] \frac{df(x)}{dx} + (x-λ)f(x) = 0 [/tex] [tex]\text{Integrating factor is }e^{\int (x-λ)dx}=e^{\frac{1}{2} x^2-λx}[/tex]
[tex]e^{\frac{1}{2} x^2-λx}f(x)=constant[/tex]

I've done a similar thing for parts a), b), and c) but I'm not sure what to do with this or if it even helps. For d) I've tried to work out the eigenfunctions but get to mess and didn't really want to continue down the route I was going without knowing if this was useful or not.

[tex]\text{b) leads to } e^{ix(\frac{x^2}{3}-λ)}f(x)=constant[/tex]
[tex]\text{c) leads to } x^{iλ}f(x)=constant[/tex]
[tex]\text{d) } ix \frac{d^3f(x)}{dx^3}=λf(x) [/tex]
[tex] x \frac{d^3f(x)}{dx^3}+iλf(x)=0 [/tex]
[tex]\text{let }x=f(t)[/tex]
[tex]\frac{d^3f(x)}{dx^3}=\frac{d^3f(x)}{dt^3}\frac{d^3t}{dx^3} [/tex]
[tex]\text{want }\frac{d^3t}{dx^3}=\frac{1}{x}[/tex]
[tex]\text{(After integrating 3 times }t=\frac{1}{2}(x^2(ln(x)-\frac{3}{2})) [/tex]

This is were I decided not to continue until I knew whether or not I was actually doing anything right.

Thanks in advance for any help.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
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Homework Statement



Consider the set of functions [itex] {f(x)} [/itex] of the real variable [itex] x [/itex] de fined on the interval [itex] -\infty< x < \infty [/itex] that go
to zero faster than [itex] 1/x [/itex] for [itex]x\rightarrow ±\infty[/itex] , i.e.,
[tex]
\lim_{n\rightarrow ±\infty} {xf(x)}=0
[/tex]
For unit weight function, determine which of the following linear operators is Hermitian when acting upon [itex] {f(x)} [/itex]:
[tex] (a) \frac{d}{dx} + x[/tex] [tex](b) -i \frac{d}{dx}+x^2 [/tex][tex](c) ix \frac{d}{dx} [/tex][tex](d) ix \frac{d^3}{dx^3} .[/tex]


Homework Equations



[itex] Hf(x)=λf(x) [/itex] has real values of [itex] λ [/itex] where [itex] H [/itex] is a Hermitian operator and [itex] λ [/itex] are it's eigenvalues



The Attempt at a Solution



[tex] a) \frac{df(x)}{dx} + xf(x) = λf(x) [/tex][tex] \frac{df(x)}{dx} + (x-λ)f(x) = 0 [/tex] [tex]\text{Integrating factor is }e^{\int (x-λ)dx}=e^{\frac{1}{2} x^2-λx}[/tex]
[tex]e^{\frac{1}{2} x^2-λx}f(x)=constant[/tex]

I've done a similar thing for parts a), b), and c) but I'm not sure what to do with this or if it even helps. For d) I've tried to work out the eigenfunctions but get to mess and didn't really want to continue down the route I was going without knowing if this was useful or not.

[tex]\text{b) leads to } e^{ix(\frac{x^2}{3}-λ)}f(x)=constant[/tex]
[tex]\text{c) leads to } x^{iλ}f(x)=constant[/tex]
[tex]\text{d) } ix \frac{d^3f(x)}{dx^3}=λf(x) [/tex]
[tex] x \frac{d^3f(x)}{dx^3}+iλf(x)=0 [/tex]
[tex]\text{let }x=f(t)[/tex]
[tex]\frac{d^3f(x)}{dx^3}=\frac{d^3f(x)}{dt^3}\frac{d^3t}{dx^3} [/tex]
[tex]\text{want }\frac{d^3t}{dx^3}=\frac{1}{x}[/tex]
[tex]\text{(After integrating 3 times }t=\frac{1}{2}(x^2(ln(x)-\frac{3}{2})) [/tex]

This is were I decided not to continue until I knew whether or not I was actually doing anything right.

Thanks in advance for any help.
No, that's really not the right way to go about it. You've likely defined an inner product ##<f,g>= \int_{-\infty}^\infty f^*(x) g(x) dx ##. In terms of that inner product H is Hermitian if <f,Hg>=<Hf,g> for any two functions f and g. Does that sound familiar? Checking that is the way to check if an operator is Hermitian. Start with the two parts of the first one. Is x Hermitian? That's pretty easy. Now is d/dx Hermitian? That's a little harder. Try looking at an integration by parts.
 
  • #3
2
0
Oh, thank you I knew I must've been doing something wrong.
 

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