Question

(2) Consider the 'square' of the derivative operator in one dimension, ie., D"--2 ie. Find all solutions of without any restrictions on , ie.,-oo< (2b) Are there limitations on a? (e.g., any complex number, any real number, only positive real number, integers only, etc.). In mathematis? In physics, when the operator is meant to represent an observable? (2c) Can you find a solution to (4) which is not a solution to DIf(x) Vaf(x)? Hint: try trig functions! Note the 'philosophical' points: (i) (4) is an eigenvalue problem; (ii) we are looking for functions which have a simple relationship between themselves and their derivatives (exponentials and sin/cos!); (ii) we certainly can have complex-valued linear coefficients when mixing independent solutions f(x) and f2(x).

Answer 1

Ans.

(a) we have to solve the second order diffrential equation.

......eq(1)

(a) we have to solve the second order diffrential equation. D^2(f(x)) = d^2f/dx^2 = alpha f(x) - (1) let the solution of this equation be f(x) = e^ax - (2) where a is an constant for being the solution of the diffrential equation it must satisfy the given diffrential equation put eq(1) in eq(2) we get d^2e^ax/dx^2 = alpha e^ax a^2 e^ax = alpha e^ax Rightarrow a^2 = alpha or Rightarrow a = squareroot a put value of a in eq(2) hence, the solution of given differential equation is f(x) = e^squareroot ax - (3) (b) Mathematically can have any value (i.e., positive, negative, real, imaginary), but in physics when this operator repersent an observable the value of must be real otherwise eigenvalues become imaginary. but in physical world we can't measure an imaginary observable (i.e., we can't have imaginary position of a particle, imaginary momentum or energy etc all we can measure is real quantities). (c) for diffrential equation

let the solution of this equation be

$\large f(x)=e^{ax}$ ....eq(2)    where a is an constant

for being the solution of the diffrential equation it must satisfy the given diffrential equation

put eq(1) in eq(2) we get

$\large \frac{\mathrm{d^{2}}e^{ax} }{\mathrm{d} x^{2}}=\alpha \ e^{ax}$

$\large a^{2} e^{ax}=\alpha e^{ax}$

$\large \implies a^{2} =\alpha$

or

$\large \implies a=\sqrt{\alpha}$

put value of a in eq(2)

hence, the solution of given differential equation is

$\large f(x)=e^{\sqrt{\alpha}x}$ .......eq(3)

(b) Mathematically $\alpha$ can have any value (i.e., positive , negative , real, imaginary),

but in physics when this operator repersent an observable the value of $\alpha$ must be real otherwise eigenvalues become imaginary. but in physical world we can't measure an imaginary observable (i.e., we can't have imaginary position of a particle, imaginary momentum or energy etc all we can measure is real quantities).

(c) for diffrential equation

$\large D(f(x))=\frac{\mathrm{d} f(x)}{\mathrm{d} x}=\sqrt{\alpha}f(x)$ .....eq(4)

the solution of above equation is

$\large f(x)=e^{\sqrt{\alpha}x}$ .....eq(5)

for D operator is meant to repersent an observable

its eigenvalues must be real

hence, $\sqrt{\alpha}$ must be real and hence $\alpha$ must be positive.

let us take $\alpha =-1$

the solution of diffrential equation in eq(1) is

$\large f(x)=e^{\sqrt{-1}x}=e^{ix}$

but for $\alpha =-1$    the solution of diffrential equation in eq(4) is not define because for this case $\alpha$ must be positive.

hence,

$e^{ix}$    is solution of differential equation (1) but not a solution of differential equation (4)