- #1
Shredface
- 6
- 0
How do I show that the scalar product is time independent?
I have: [tex]\frac{d}{dt}\int\Psi^{*}_{1}(x,t)\Psi_{2}(x,t)dx = 0[/tex]
And have proceeded to take the derivatives inside the integral and using the time dependent Schrodinger eq. ending up with:
[tex]\frac{i}{\hbar}\int\left(\Psi^{*}_{1}\widehat{H}\Psi_{2}-\Psi_{2}\widehat{H}\Psi^{*}_{1}\right)dx[/tex]
Using the Hermitian properties of [tex]\widehat{H}[/tex] I then got to:
[tex]\frac{i}{\hbar}\int\left(\Psi^{*}_{1}\widehat{H}\Psi_{2} - \Psi_{1}\widehat{H}\Psi^{*}_{2}\right)dx[/tex]
Either these two terms equivalent meaning they cancel and I have my result or I've made a drastic error somewhere. Can anyone help me out?
I have: [tex]\frac{d}{dt}\int\Psi^{*}_{1}(x,t)\Psi_{2}(x,t)dx = 0[/tex]
And have proceeded to take the derivatives inside the integral and using the time dependent Schrodinger eq. ending up with:
[tex]\frac{i}{\hbar}\int\left(\Psi^{*}_{1}\widehat{H}\Psi_{2}-\Psi_{2}\widehat{H}\Psi^{*}_{1}\right)dx[/tex]
Using the Hermitian properties of [tex]\widehat{H}[/tex] I then got to:
[tex]\frac{i}{\hbar}\int\left(\Psi^{*}_{1}\widehat{H}\Psi_{2} - \Psi_{1}\widehat{H}\Psi^{*}_{2}\right)dx[/tex]
Either these two terms equivalent meaning they cancel and I have my result or I've made a drastic error somewhere. Can anyone help me out?