bei48482_FM

and so we have

KˆE

2

 (5.26)

Equation (5.25) therefore reads

iU (5.27)

Now we multiply the identity by Eq. (5.27) and obtain

iU

which is Schrödinger’s equation. Postulating Eqs. (5.23) and (5.24) is equivalent to

postulating Schrödinger’s equation.

Operators and Expectation Values

Because pand Ecan be replaced by their corresponding operators in an equation, we

can use these operators to obtain expectation values for pand E. Thus the expectation

value for pis

p 





*pˆ dx





*  dx 





* dx (5.28)

and the expectation value for Eis

E 



*Eˆ dx



i  dxi (^) 

 dx (5.29)
Both Eqs. (5.28) and (5.29) can be evaluated for any acceptable wave function (x,t).
Let us see why expectation values involving operators have to be expressed in the
form
p 


pˆ dx
The other alternatives are



pˆ dx 


() dx 


 0
since  and must be 0 at x , and



ˆpdx 


* dx
which makes no sense. In the case of algebraic quantities such as xand V(x), the order
of factors in the integrand is unimportant, but when differential operators are involved,
the correct order of factors must be observed.

x
i
i

x
i

t

t

x
i

x
i
^2 
x^2
2
2 m

t
^2
x^2
2
2 m

t
^2
x^2
2
2 m

x
i
1
2 m
pˆ^2
2 m
Kinetic-energy
operator
Quantum Mechanics 173
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