lecture04_umn

# lecture04_umn - Lecture 4 The Wave Function for a Material...

This preview shows pages 1–9. Sign up to view the full content.

Lecture 4 •The Wave Function for a Material System •The Time-independent Schrödinger Equation •What is a Wave Function? •Collapse of the Wave Function September 16 2009 Pages 31-72 McQuarrie and Simon

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Wave Function for a Material System Erwin Schrödinger, in 1926, proposed a way to meld the many quantum observations up to that point with a wavelike description of matter . He started from the classical wave equation (here for a single dimension) (4-1) where Ψ is a wave function which has an amplitude for any specification of x (position) and t (time) , and c is the velocity of the wave (wavelength times frequency). A general solution to this equation is (4-2) where C is an arbitrary multiplicative constant, i is , λ is the wavelength and ν is the frequency. " 2 # x , t ( ) x 2 = 1 c 2 2 # x , t ( ) t 2 " x , t ( ) = Ce 2 # i x \$ % & t ( ) * + , 1
Wave Function Description To verify, note that (4-3) and (4-4) and c = λν . Schrödinger decided to use the de Broglie wavelength for λ (3- 21) and to relate the frequency to the Planck energy(2-4) . " 2 # x , t ( ) x 2 = \$ 4 C % 2 & 2 e 2 i x \$ t ( ) * + , - 2 # x , t ( ) t 2 = \$ 4 C 2 2 e 2 i x \$ t ( ) * + , - = h p Ε = h ν (2-4) (3-21)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Differential equation He looked for a differential equation having the solution (4-5) where p is the momentum and E is the energy . For simplicity, we will take the arbitrary constant C as one. If we consider differentiating eq. 4-5 once with respect to time we have (4-6) or (4-7) "# x , t ( ) " t = \$ iE h e i xp \$ Et h % & ( ) * " h i #\$ x , t ( ) # t = E \$ x , t ( ) " x , t ( ) = Ce i xp # Et h \$ % & ( )
Information contained in the Wave Function (4-7) The total energy E at a given time may be expressed as a sum of kinetic and potential energy , both of which depend only on x and not on t : (4-8) where p is the momentum , m is the mass , and V is the potential energy , which may be thought of as an outside influence on the system being described by the wave function Ψ . " h i #\$ x , t ( ) # t = p 2 2 m + V x ( ) % & ( ) * \$ x , t ( ) " h i x , t ( ) t = E \$ x , t ( )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Information contained in the Wave Function Note that the idea here is to have Ψ be a function that contains information about the system contained within it . That is, we would like things like momentum and energy to be themselves determinable from the wave function . Wave function (4-5) " h i #\$ x , t ( ) # t = p 2 2 m + V x ( ) % & ( ) * \$ x , t ( ) (4-8) " x , t ( ) = e i xp # Et h \$ % & ( )
How can we determine the momentum from Ψ ? Differentiate once with respect to x: (4-9) Write this in "operator" formalism as (4-10) or (4-11) "# x , t ( ) " x = ip h e i xp \$ Et h % & ( ) * x # x , t ( ) = ip h # x , t ( ) h i x = p " x , t ( ) = e i xp # Et h \$ % & ( )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Momentum Operator We say that the operator that can be applied to the wave function in order to determine the momentum is to differentiate once with respect to x and then multiply by h-bar over i . In this case, it is straightforward to show that the square of the momentum (needed for the kinetic energy) can be derived from operating twice with the momentum operator , i.e., (4-12) If we include this result in equation 4-8 we have (4-13) This is the one-dimensional, time-dependent Schrödinger equation .
This is the end of the preview. Sign up to access the rest of the document.

## lecture04_umn - Lecture 4 The Wave Function for a Material...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document