(英文)量子力学-薛定谔方程

( x , y , z , t ) ( x , y , z ) F ( t )
Then
i ( x , y, z )F ( t ) H ( x , y, z ) ( x , y, z )F ( t ) t
i ( x, y, z ) F ( t ) F ( t ) H ( x , y, z ) ( x, y, z ) t
depends only on t
Can only be true for any x, y, z, t if both sides equal a constant. Changing t on the left doesn’t change the value on the right. Changing x, y, z on right doesn’t change value on left. Equal constant
Total wavefunction is normalized if time independent part is normalized.
Expectation value of time independent operator S.
* iEt/ S S * S E e i E t / d E S E d E e
i
wavefunction
( x , y , z , t ) H ( x , y , z , t )( x , y, z , t ) t
If the energy is independent of time
Try solution
H ( x, y, z )
product of spatial function and time function
kinetic potential energy energy
2 2 H V (x) 2 2m x
2 2 H V ( x, y, zwenku.baidu.com) 2m
one dimension
three dimensions
recall
2
p i
x
2 2 2 2 2 x y z2
Schrö dinger Representation – Schrö dinger Equation
Time dependent Schrö dinger Equation
i
( x , y , z , t ) H ( x , y , z , t )( x , y, z , t ) t
i
dF dt E H F
i
dF dt E H F
Both sides equal a constant, E.
H ( x, y, z ) ( x, y, z ) E ( x, y, z )
Energy eigenvalue problem – time independent Schrö dinger Equation H is energy operator.
Total wavefunction
E ( x, y, z, t ) E ( x, y, z )e i E t /
Normalization
E – energy (observable) that labels state.
* i E t / i E t / * E E * d e e d E E E E E E d
independent of t independent of x, y, z divide through by
F
i
dF ( t ) dt H ( x , y , z ) ( x , y , z ) F (t ) ( x, y, z )
depends only on x, y, z
p2 V 2m
Developed through analogy to Maxwell’s equations and knowledge of the Bohr model of the H atom.
H classical
Hamiltonian Q.M.
Sum of kinetic energy and potential energy.
The potential, V, makes one problem different form another H atom, harmonic oscillator.
Getting the Time Independent Schrö dinger Equation
( x , y , z , t )
i dF ( t ) dt E F (t )
i
dF ( t ) E F (t ) dt
dF ( t ) i E dt . F (t )
ln F iEt C
Integrate both sides
F (t ) e i E t / e it
Time dependent part of wavefunction for time independent Hamiltonian. Time dependent phase factor used in wave packet problem.
S does not depend on t, ei Et / can be brought to other side of S.
Operate on get back times a number.
’s are energy eigenkets; eigenfunctions; wavefunctions.
E Energy Eigenvalues Observable values of energy
Time Dependent Equation (H time independent)