exercícios com resposta sobre mecânica quântica

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CHEM-E4110

Quantum mechanics and Spectroscopy

Exercise II

Timo Weckman, [email protected]

Done during the exercise session

During this exercise we use Maple to study basic quantum systems.

Exercise 1 Hyperbolic secant potential

Consider the Schrdinger equation with a potential

V(x) = 2a2

m sech2(ax)

where a is a positive constant and sech is the hyperbolic secant, sech(x) =1

cosh(x).

a) What does the potential look like?

b) Check that the potential has the ground state

0(x) =A sech(ax)

Normalise the wave function and plot it. What is the energy of the groundstate?

Exercise 2 Non-physical solutions to harmonic oscillator

In this exercise you solve the ground state harmonic oscillator numericallyusing dsolve.

a) First show that the Schrdinger equation for harmonic oscillator

2

2m

d2

dx2(x) +

1

2m2x2(x) =E (x)

can be written as

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d2

d2() +2() =()

by substitution of variables, =

m

xand = 2E

.

b) Write the Schrdinger equation into Maple as in the previous exercisesand set parameter type=numeric in dsolve. The value of you get fromconsidering what is the ground state energy of the harmonic oscillator. Asinitial conditions for the ground state you can use (0) = 1, (0) = 0(why?).Plot your solution using plots[odeplot] (a recommended plotting range isx= 5..5 as the numerical solution becomes unstable).c) Solve the ground state wave function with an epsilon slightly less (0.99) and

slightly more (1.01) than the ground state value. How do your plots differ?What does this mean?

Exercise 3 Superposition of states

Suppose the harmonic oscillator is in a superpositionof states 1 and 2 (seeanalytical form for the two functions at the end of the exercise sheet):

(x) = 1

2(0+1)

a) Check that the wave function is properly normalized.b) Is the state an eigenstate of the Hamiltonian H?

c) What is the total energy of the system?

d) Consider the time-evolution of the system, i.e. multiply each eigenstate i

with the time-factor eiEit

(x, t) = 1

2

0 e

iE0t

+1 eiE1t

Is the superposition state a stationary state of the Hamiltonian? Is the total

energy conserved? Is the total density conserved?e) What does the density function|(x)|2 look like as a function of time? Forthis you can use the plots[animate]-function of Maple.

Homework

These exercises are to be returned for the first time (via email) before the nextexercise session. You can ask for help any time by email or come tomy office C242b.

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Exercise 1 The wave function

Answer these questions briefly (only few sentences):a) What conditions are required of the wave function for it to represent aphysical state?

b) What is required of a quantum mechanical operator in order for it to be anobservable?

c) When is it possible to separate time-dependent Schrdinger equation intothe time-independent form?

d) What are so-called stationary states?

Exercise 2 Hyperbolic secant potential revisited

Consider the Schrdinger equation with a potential

V(x) = 2a2

m sech2(ax)

where a is a positive constant and sech is the hyperbolic secant, sech(x) =1

cosh(x).

The exact solution to the ground state is

0(x) =A sech(ax)

Approximate the exact solution using a normalized trial function (x),

(x) =

2b

14

ebx2

Plot the trial function with the exact ground state wave function using thedifferent values for b= 0.1, 0.25, 1 (assume value a= 1 for the calculations).

a) What is value of the overlap integral

0(x)(x)dx

with different value ofb? What does the integral mean?

b) Calculate the expectation value of the total energy using a trial function.How does the expectation value compare to the exact ground state energy withdifferent values ofb? What trend do you see?

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Harmonic oscillator wave functions

Eigenfunctions for few of the lowest eigenstates of the harmonic oscillator:

0(x) =

14

ex

2

2

1(x) =

43

14

xex

2

2

2(x) =

4

14

(2x2 1)ex2

2

3(x) =

3

9

14

(2x3 3x)ex2

2

4

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