Aula 8

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UFABC - Física Quântica - Curso 2017.3 Prof. Germán Lugones

Aula 8

A equação de Schrödinger

1

6-1 The Schrödinger Equation in One Dimension 231

then use this relation to work backward and see how the wave equation for electrons must differ from Equation 6-1. The total energy (nonrelativistic) of a particle of mass m is

E p

2

2m V 6-4

where V is the potential energy. Substituting the de Broglie relations in Equation 6-4, we obtain

2_k2

2m V 6-5

This differs from Equation 6-2 for a photon because it contains the potential energy V and because the angular frequency does not vary linearly with k. Note that we get a factor of when we differentiate a harmonic wave function with respect to time and a factor of k when we differentiate with respect to position. We expect, therefore, that the wave equation that applies to electrons will relate the first time derivative to the

second space derivative and will also involve the potential energy of the electron.

Finally, we require that the wave equation for electrons will be a differential equation that is linear in the wave function (x, t). This ensures that, if 1(x, t) and 2(x, t) are both solutions of the wave equation for the same potential energy, then any arbitrary linear combination of these solutions is also a solution—that is, (x, t)

a1 1(x, t) a2 2(x, t) is a solution, with a1 and a2 being arbitrary constants. Such a combination is called linear because both 1(x, t) and 2(x, t) appear only to the first power. Linearity guarantees that the wave functions will add together to produce con-structive and decon-structive interference, which we have seen to be a characteristic of matter waves as well as all other wave phenomena. Note in particular that (1) the lin-earity requirement means that every term in the wave equation must be linear in (x, t) and (2) that any derivative of (x, t) is linear in (x, t).4

Erwin Schrödinger. [Courtesy of the Niels Bohr Library, American Institute of Physics.]

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Na primeira parte do curso, introduzimos a dualidade onda-partícula.

Usando as relações de Planck e de De Broglie, conseguimos explicar experimentos tais como:

- corpo negro

- efeito fotoelétrico - efeito Compton - átomo de Bohr - etc....

Para analisar situações mais complexas precisamos de uma teoria mais elaborada:

- por exemplo, o postulado de de Broglie não diz como é a onda que representa uma partícula cujo momento não é constante.

- também não diz como se propaga essa onda.

2

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A equação fundamental da teoria quântica, foi introduzida em 1926 por Schrödinger.

- essa equação, não pode ser demonstrada: ela é um dos postulados ou axiomas da teoria.

- para determinar se a equação de Schrödinger representa o comportamento dos sistemas quânticos, devemos comparar as predições da equação com os experimentos.

(4)

4

Vamos postular que a função de onda Ψ(x, t) de uma partícula de massa m, movendo-se em uma dimensão sob a influência de um potencial V(x, t), obedece a equação de Schrödinger:

- x = coordenada espacial. - t = coordenada temporal.

- Ψ(x,t) = função de onda da partícula considerada. Depende de x e t.

- V(x, t) = função de energia potencial onde se encontra imersa a partícula considerada. Por exemplo, potencial elétrico de um núcleo, potencial elétrico de várias cargas, potencial molecular de Lennard-Jones, etc…

- i = número imaginário = (–1)1/2

- A equação de Schrödinger é uma equação diferencial de segunda ordem em derivadas parciais que permite obter a função de onda.

232 Chapter 6 The Schrödinger Equation

The Schrödinger Equation

We are now ready to postulate the Schrödinger equation for a particle of mass m. In

one dimension, it has the form

2

2m

2

_{x, t}

x

2

V x, t

x, t

i

x, t

t

6-6

We will now show that this equation is satisfied by a harmonic wave function in the

special case of a free particle, one on which no net force acts, so that the potential

energy is constant, V(x, t) V

₀

. First note that a function of the form cos(kx t)

does not satisfy this equation because differentiation with respect to time changes the

cosine to a sine but the second derivative with respect to x gives back a cosine.

Simi-lar reasoning rules out the form sin(kx t). However, the exponential form of the

harmonic wave function does satisfy the equation. Let

x, t

Ae

i kx t

A cos kx

t

i sin kx

t

6-7

where A is a constant. Then

t

i A e

i kx t

_i

and

2

x

2

ik

2

_{A e}

i kx t

_k

2

Substituting these derivatives into the Schrödinger equation with V(x, t) V

₀

gives

2

2m

k

2

_V

0

i

or

2

_k

2

2m

V

0

which is Equation 6-5.

An important difference between the Schrödinger equation and the classical

wave equation is the explicit appearance

5

of the imaginary number i

1

1 2

. The

wave functions that satisfy the Schrödinger equation are not necessarily real, as we

see from the case of the free-particle wave function of Equation 6-7. Evidently the

wave function (x, t) that solves the Schrödinger equation is not a directly

measur-able function like the classical wave function y(x, t) since measurements always yield

real numbers. However, as we discussed in Section 5-4, the probability of finding the

electron in some region dx is certainly measurable, just as is the probability that a

flipped coin will turn up heads. The probability P(x) dx that the electron will be found

in the volume dx was defined by Equation 5-23 to be equal to

2

dx. This probabilistic

interpretation of was developed by Max Born and was recognized, over the early

and formidable objections of both Schrödinger and Einstein, as the appropriate way

of relating solutions of the Schrödinger equation to the results of physical

measure-ments. The probability that an electron is in the region dx, a real number, can be

mea-sured by counting the fraction of time it is found there in a very large number of

identical trials. In recognition of the complex nature of

(x, t), we must modify

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5

Um número complexo qualquer pode ser representado por:

z = x + i y

onde x e y são números reais:

x é a parte real de z: x = Re(z)

y é a parte imaginária de z: y = Im(z)

O símbolo i possui a seguinte propriedade: i

2

= –1

(6)

6

Números complexos podem ser representados no plano de Argand,

onde os eixos horizontal e vertical representam os eixos real e

imaginário, respectivamente:

Plano de Argand

N´umeros complexos

Aula 10 4 / 22

■

N´

umeros complexos podem ser representados no plano de Argand, onde os

eixos horizontal e vertical representam os eixos real e imagin´ario,

respectivamente:

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7

O complexo conjugado de um número complexo z=x+iy é definido

por:

z =(x+yi)* = x−yi*

 

z é a reflexão de z em relação ao eixo real no plano de Argand.*

O complexo conjugado

N´umeros complexos

Aula 10 6 / 22

O complexo conjugago de um n´

umero complexo

z = x + iy ´e definido por

z

∗

= (x + yi)

∗

= x − yi

z

∗

é a reflexão de z em rela¸cão ao eixo real no plano

de Argand.

z = x + iy

z∗ = _x − iy

Propriedades

■

Temos que i

∗

= −i, enquanto que x

∗

= x e y

∗

= y, pois x e y s˜ao n´

umeros

reais.

■

(z

₁

+ z

₂

)

∗

= z

₁∗

+ z

₂∗

■

(z

₁

z

₂

)

∗

= z

₁∗

z

₂∗

(8)

8

O módulo ou valor absoluto, |z|, de um número complexo z é a

distância do ponto z até a origem no plano de Argand:

z = (x

2

+ y

2

)

1/2

Observa-se que:

zz = (x + iy)(x – iy) = x*

2

+ y

2

Portanto:

zz = |z|*

2

M´

odulo ou valor absoluto

N´umeros complexos

Aula 10 7 / 22

O m´

odulo

_{ou valor absoluto, |z|, de um n´umero}

complexo z é a distância do ponto z até a origem

no plano de Argand:

|z| =

!

x

2

+ y

2 z = x + iy x |z|= √ x2 + y2 yi

Observa-se que

zz

∗

= (x + yi)(x − yi) = x

2

+ y

2

Logo, zz

∗

= |z|

2

, ou |z| =

√

zz

∗

.

■

Em termos do m´odulo, o quociente ´e dado por

z

₁

z

₂

=

z

₁

z

₂

z

∗ 2

z

∗ 2

=

z

1

z

∗ 2

|z

2

|

2

(9)

9

Forma polar:

De acordo com a figura,

podemos representar um

número complexo z=x+iy

usando:

x = r cos

𝜃 y=r sin𝜃

Portanto, temos:

z = r (cos

𝜃 + i sin𝜃)

Forma polar

N´umeros complexos

Aula 10 10 / 22

Podemos representar um ponto z = x+yi no plano

de Argand em termos de r e θ, conforme figura ao

lado. Como

x = r cos θ

e y = r sen θ

temos que z = r(cos θ + i sen θ)

z = x + yi

x yi

Temos que

■

r = |z| =

!

x

2

+ y

2

■

θ ´e o argumento de z, θ = arg(z), com

tg θ =

y

x

(10)

10

Fórmula de Euler:

Para a função real de variável real f(x)=e

x

, a sua série de Taylor em

torno do ponto x

0

= 0 é dada por

Vamos assumir que a expansão acima também se aplica a uma função

de variável complexa; logo:

Para o caso particular em que z=iy, temos:

ex = 1 X n=0 xn n! = 1 + x + x2 2! + x3 3! + · · · ez = 1 X n=0 zn n! = 1 + z + z2 2! + z3 3! + · · ·

e

iy

=

1

X

n=0

(iy)

n

n!

= 1 + (iy) +

(iy)

2

2!

+

(iy)

3

3!

+

· · ·

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11

Na expressão anterior podemos usar:

i

2

= –1, i

3

= i

2

i = –i, i

4

=1, i

5

= i, …

Logo, temos

Na expressão anterior, identificamos o primeiro termo entre

parênteses como sendo a série de Taylor de cos(y) e o segundo como

sen(y). Portanto,

que é a expressão conhecida como fórmula de Euler.

e

iy

= 1 + iy

y

2

2!

i

y

3

3!

+

y

4

4!

+ i

y

5

5!

· · ·

=

✓

1 y

2

2!

+

y

4

4!

· · ·

◆ + i

✓

y

3

3!

+

y

5

5!

· · ·

◆ e

iy

= cos(y) + i sin(y)

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12

A equação de Schrödinger apresenta algumas propriedades importantes: Propriedade 1: É uma equação linear em ψ(x,t):

Sejam ψ1(x,t) e ψ2(x,t) duas soluções diferentes da equação para uma dada

energia potencial V. Logo:

ψ_{(x,t) = C}₁_ψ₁_{(x,t) + C}₂_ψ₂_(x,t)

também é uma solução da equação de Schrödinger para quaisquer constantes C1 e C2 .

A linearidade da equação de Schrödinger garante que podemos obter interferências construtivas e destrutivas aos somarmos duas ondas.

(13)

13

Propriedade 2: para uma onda senoidal, a equação de Schrödinger é consistente com a relação:

onde:

- E é a energia total da partícula,

- p2/(2m) é a energia cinética da partícula,

- V é a energia potencial da mesma. Demonstração: consideremos uma onda senoidal:

onde A é uma constante.

6-1 The Schrödinger Equation in One Dimension 231

then use this relation to work backward and see how the wave equation for electrons must differ from Equation 6-1. The total energy (nonrelativistic) of a particle of mass m is

E p

2

2m V 6-4

where V is the potential energy. Substituting the de Broglie relations in Equation 6-4, we obtain

2_k2

2m V 6-5

This differs from Equation 6-2 for a photon because it contains the potential energy V and because the angular frequency does not vary linearly with k. Note that we get a factor of when we differentiate a harmonic wave function with respect to time and a factor of k when we differentiate with respect to position. We expect, therefore, that the wave equation that applies to electrons will relate the first time derivative to the

second space derivative and will also involve the potential energy of the electron.

Finally, we require that the wave equation for electrons will be a differential equation that is linear in the wave function (x, t). This ensures that, if ₁(x, t) and

2(x, t) are both solutions of the wave equation for the same potential energy, then any

arbitrary linear combination of these solutions is also a solution—that is, (x, t)

a₁ ₁(x, t) a₂ ₂(x, t) is a solution, with a₁ and a₂ being arbitrary constants. Such a combination is called linear because both ₁(x, t) and ₂(x, t) appear only to the first power. Linearity guarantees that the wave functions will add together to produce con-structive and decon-structive interference, which we have seen to be a characteristic of matter waves as well as all other wave phenomena. Note in particular that (1) the lin-earity requirement means that every term in the wave equation must be linear in (x, t) and (2) that any derivative of (x, t) is linear in (x, t).4

Erwin Schrödinger. [Courtesy of the Niels Bohr Library, American Institute of Physics.]

TIPLER_06_229-276hr.indd 231 8/22/11 11:57 AM

232

Chapter 6 The Schrödinger Equation

The Schrödinger Equation

We are now ready to postulate the Schrödinger equation for a particle of mass m. In

one dimension, it has the form

2

2m

2

_{x, t}

x

2

V x, t

x, t

i

x, t

t

6-6

We will now show that this equation is satisfied by a harmonic wave function in the

special case of a free particle, one on which no net force acts, so that the potential

energy is constant, V(x, t) V

₀

. First note that a function of the form cos(kx t)

does not satisfy this equation because differentiation with respect to time changes the

cosine to a sine but the second derivative with respect to x gives back a cosine.

Simi-lar reasoning rules out the form sin(kx t). However, the exponential form of the

harmonic wave function does satisfy the equation. Let

x, t

Ae

i kx t

A cos kx

t

i sin kx

t

6-7

where A is a constant. Then

t

i A e

i kx t

_i

and

2

x

2

ik

2

_{A e}

i kx t

_k

2

Substituting these derivatives into the Schrödinger equation with V(x, t) V

₀

gives

2

2m

k

2

_V

0

i

or

2

_k

2

2m

V

0

which is Equation 6-5.

An important difference between the Schrödinger equation and the classical

wave equation is the explicit appearance

5

of the imaginary number i

1

1 2

. The

wave functions that satisfy the Schrödinger equation are not necessarily real, as we

see from the case of the free-particle wave function of Equation 6-7. Evidently the

wave function (x, t) that solves the Schrödinger equation is not a directly

measur-able function like the classical wave function y(x, t) since measurements always yield

real numbers. However, as we discussed in Section 5-4, the probability of finding the

electron in some region dx is certainly measurable, just as is the probability that a

flipped coin will turn up heads. The probability P(x) dx that the electron will be found

in the volume dx was defined by Equation 5-23 to be equal to

2

dx. This probabilistic

interpretation of was developed by Max Born and was recognized, over the early

and formidable objections of both Schrödinger and Einstein, as the appropriate way

of relating solutions of the Schrödinger equation to the results of physical

measure-ments. The probability that an electron is in the region dx, a real number, can be

mea-sured by counting the fraction of time it is found there in a very large number of

identical trials. In recognition of the complex nature of

(x, t), we must modify

(14)

14

Usando a função de onda senoidal:

calculamos as duas derivadas presentes na equação de Schrödinger:

Substituindo essas derivadas na equação de Schrödinger, e usando V0 =

V(x,t) = constante, temos:

232 The Schrödinger Equation

We are now ready to postulate the Schrödinger equation for a particle of mass m. In

one dimension, it has the form

2

2m

2

_{x, t}

x

2

V x, t

x, t

i

x, t

t

6-6

We will now show that this equation is satisfied by a harmonic wave function in the

special case of a free particle, one on which no net force acts, so that the potential

energy is constant, V(x, t) V

₀

. First note that a function of the form cos(kx t)

does not satisfy this equation because differentiation with respect to time changes the

cosine to a sine but the second derivative with respect to x gives back a cosine.

Simi-lar reasoning rules out the form sin(kx t). However, the exponential form of the

harmonic wave function does satisfy the equation. Let

x, t

Ae

i kx t

A cos kx

t

i sin kx

t

6-7

where A is a constant. Then

t

i A e

i kx t

_i

and

2

x

2

ik

2

_{A e}

i kx t

_k

2

Substituting these derivatives into the Schrödinger equation with V(x, t) V

₀

gives

2

2m

k

2

_V

0

i

or

2

_k

2

2m

V

0

which is Equation 6-5.

An important difference between the Schrödinger equation and the classical

wave equation is the explicit appearance

5

of the imaginary number i

1

1 2

. The

wave functions that satisfy the Schrödinger equation are not necessarily real, as we

see from the case of the free-particle wave function of Equation 6-7. Evidently the

wave function (x, t) that solves the Schrödinger equation is not a directly

measur-able function like the classical wave function y(x, t) since measurements always yield

real numbers. However, as we discussed in Section 5-4, the probability of finding the

electron in some region dx is certainly measurable, just as is the probability that a

flipped coin will turn up heads. The probability P(x) dx that the electron will be found

in the volume dx was defined by Equation 5-23 to be equal to

2

dx. This probabilistic

interpretation of was developed by Max Born and was recognized, over the early

and formidable objections of both Schrödinger and Einstein, as the appropriate way

of relating solutions of the Schrödinger equation to the results of physical

measure-ments. The probability that an electron is in the region dx, a real number, can be

mea-sured by counting the fraction of time it is found there in a very large number of

identical trials. In recognition of the complex nature of

(x, t), we must modify

TIPLER_06_229-276hr.indd 232 8/22/11 11:57 AM

232 Chapter 6 The Schrödinger Equation

The Schrödinger Equation

We are now ready to postulate the Schrödinger equation for a particle of mass m. In one dimension, it has the form

2 2m 2 _{x, t} x2 V x, t x, t i x, t t 6-6

We will now show that this equation is satisfied by a harmonic wave function in the special case of a free particle, one on which no net force acts, so that the potential energy is constant, V(x, t) V₀. First note that a function of the form cos(kx t) does not satisfy this equation because differentiation with respect to time changes the cosine to a sine but the second derivative with respect to x gives back a cosine. Simi-lar reasoning rules out the form sin(kx t). However, the exponential form of the harmonic wave function does satisfy the equation. Let

x, t Aei kx t

A cos kx t i sin kx t 6-7

where A is a constant. Then

t i A e i kx t _i and 2 x2 ik 2_{A e}i kx t _k2

Substituting these derivatives into the Schrödinger equation with V(x, t) V₀ gives

2 2m k 2 _V 0 i i or 2_k2 2m V0 which is Equation 6-5.

An important difference between the Schrödinger equation and the classical wave equation is the explicit appearance5 of the imaginary number i 1 1 2. The wave functions that satisfy the Schrödinger equation are not necessarily real, as we see from the case of the free-particle wave function of Equation 6-7. Evidently the wave function (x, t) that solves the Schrödinger equation is not a directly measur-able function like the classical wave function y(x, t) since measurements always yield real numbers. However, as we discussed in Section 5-4, the probability of finding the electron in some region dx is certainly measurable, just as is the probability that a flipped coin will turn up heads. The probability P(x) dx that the electron will be found in the volume dx was defined by Equation 5-23 to be equal to 2dx. This probabilistic

interpretation of was developed by Max Born and was recognized, over the early and formidable objections of both Schrödinger and Einstein, as the appropriate way of relating solutions of the Schrödinger equation to the results of physical measure-ments. The probability that an electron is in the region dx, a real number, can be mea-sured by counting the fraction of time it is found there in a very large number of identical trials. In recognition of the complex nature of (x, t), we must modify

TIPLER_06_229-276hr.indd 232 8/22/11 11:57 AM

232 The Schrödinger Equation

We are now ready to postulate the Schrödinger equation for a particle of mass m. In

one dimension, it has the form

2

2m

2

_{x, t}

x

2

V x, t

x, t

i

x, t

t

6-6

We will now show that this equation is satisfied by a harmonic wave function in the

special case of a free particle, one on which no net force acts, so that the potential

energy is constant, V(x, t) V

₀

. First note that a function of the form cos(kx t)

does not satisfy this equation because differentiation with respect to time changes the

cosine to a sine but the second derivative with respect to x gives back a cosine.

Simi-lar reasoning rules out the form sin(kx t). However, the exponential form of the

harmonic wave function does satisfy the equation. Let

x, t

Ae

i kx t

A cos kx

t

i sin kx

t

6-7

where A is a constant. Then

t

i A e

i kx t

_i

and

2

x

2

ik

2

_{A e}

i kx t

_k

2

Substituting these derivatives into the Schrödinger equation with V(x, t) V

₀

gives

2

2m

k

2

_V

0

i

or

2

_k

2

2m

V

0

which is Equation 6-5.

An important difference between the Schrödinger equation and the classical

wave equation is the explicit appearance

5

of the imaginary number i

1

1 2

. The

wave functions that satisfy the Schrödinger equation are not necessarily real, as we

see from the case of the free-particle wave function of Equation 6-7. Evidently the

wave function (x, t) that solves the Schrödinger equation is not a directly

measur-able function like the classical wave function y(x, t) since measurements always yield

real numbers. However, as we discussed in Section 5-4, the probability of finding the

electron in some region dx is certainly measurable, just as is the probability that a

flipped coin will turn up heads. The probability P(x) dx that the electron will be found

in the volume dx was defined by Equation 5-23 to be equal to

2

dx. This probabilistic

interpretation of was developed by Max Born and was recognized, over the early

and formidable objections of both Schrödinger and Einstein, as the appropriate way

of relating solutions of the Schrödinger equation to the results of physical

measure-ments. The probability that an electron is in the region dx, a real number, can be

mea-sured by counting the fraction of time it is found there in a very large number of

identical trials. In recognition of the complex nature of

(x, t), we must modify

TIPLER_06_229-276hr.indd 232 8/22/11 11:57 AM

The Schrödinger Equation

We are now ready to postulate the Schrödinger equation for a particle of mass m. In one dimension, it has the form

2 2m 2 _{x, t} x2 V x, t x, t i x, t t 6-6

We will now show that this equation is satisfied by a harmonic wave function in the special case of a free particle, one on which no net force acts, so that the potential energy is constant, V(x, t) V₀. First note that a function of the form cos(kx t) does not satisfy this equation because differentiation with respect to time changes the cosine to a sine but the second derivative with respect to x gives back a cosine. Simi-lar reasoning rules out the form sin(kx t). However, the exponential form of the harmonic wave function does satisfy the equation. Let

x, t Aei kx t

A cos kx t i sin kx t 6-7

where A is a constant. Then

t i A e i kx t _i and 2 x2 ik 2_{A e}i kx t _k2

Substituting these derivatives into the Schrödinger equation with V(x, t) V₀ gives

2 2m k 2 _V 0 i i or 2_k2 2m V0 which is Equation 6-5.

An important difference between the Schrödinger equation and the classical wave equation is the explicit appearance5 of the imaginary number i 1 1 2. The wave functions that satisfy the Schrödinger equation are not necessarily real, as we see from the case of the free-particle wave function of Equation 6-7. Evidently the wave function (x, t) that solves the Schrödinger equation is not a directly measur-able function like the classical wave function y(x, t) since measurements always yield real numbers. However, as we discussed in Section 5-4, the probability of finding the electron in some region dx is certainly measurable, just as is the probability that a flipped coin will turn up heads. The probability P(x) dx that the electron will be found in the volume dx was defined by Equation 5-23 to be equal to 2dx. This probabilistic

interpretation of was developed by Max Born and was recognized, over the early and formidable objections of both Schrödinger and Einstein, as the appropriate way of relating solutions of the Schrödinger equation to the results of physical measure-ments. The probability that an electron is in the region dx, a real number, can be mea-sured by counting the fraction of time it is found there in a very large number of identical trials. In recognition of the complex nature of (x, t), we must modify

(15)

15

Simplificado, obtemos:

como queríamos demonstrar.

232 The Schrödinger Equation

We are now ready to postulate the Schrödinger equation for a particle of mass m. In

one dimension, it has the form

2

2m

2

_{x, t}

x

2

V x, t

x, t

i

x, t

t

6-6

We will now show that this equation is satisfied by a harmonic wave function in the

special case of a free particle, one on which no net force acts, so that the potential

energy is constant, V(x, t) V

₀

. First note that a function of the form cos(kx t)

does not satisfy this equation because differentiation with respect to time changes the

cosine to a sine but the second derivative with respect to x gives back a cosine.

Simi-lar reasoning rules out the form sin(kx t). However, the exponential form of the

harmonic wave function does satisfy the equation. Let

x, t

Ae

i kx t

A cos kx

t

i sin kx

t

6-7

where A is a constant. Then

t

i A e

i kx t

_i

and

2

x

2

ik

2

_{A e}

i kx t

_k

2

Substituting these derivatives into the Schrödinger equation with V(x, t) V

₀

gives

2

2m

k

2

_V

0

i

or

2

_k

2

2m

V

0

which is Equation 6-5.

An important difference between the Schrödinger equation and the classical

wave equation is the explicit appearance

5

of the imaginary number i

1

1 2

. The

wave functions that satisfy the Schrödinger equation are not necessarily real, as we

see from the case of the free-particle wave function of Equation 6-7. Evidently the

wave function (x, t) that solves the Schrödinger equation is not a directly

measur-able function like the classical wave function y(x, t) since measurements always yield

real numbers. However, as we discussed in Section 5-4, the probability of finding the

electron in some region dx is certainly measurable, just as is the probability that a

flipped coin will turn up heads. The probability P(x) dx that the electron will be found

in the volume dx was defined by Equation 5-23 to be equal to

2

dx. This probabilistic

interpretation of was developed by Max Born and was recognized, over the early

and formidable objections of both Schrödinger and Einstein, as the appropriate way

of relating solutions of the Schrödinger equation to the results of physical

measure-ments. The probability that an electron is in the region dx, a real number, can be

mea-sured by counting the fraction of time it is found there in a very large number of

identical trials. In recognition of the complex nature of

(x, t), we must modify

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16

Propriedade 3: A equação de Schrödinger não é válida para partículas que se movem com velocidades relativísticas, i.e. não conseguimos chegar na relação E = (p2c2 + m2c4)1/2 + V.

Em 1927, Klein e Gordon desenvolveram uma teoria relativística para a Mecânica Quântica → serve para bósons de spin zero.

Em 1928, Dirac desenvolveu outra teoria relativística para a Mecânica Quântica → serve para férmions (elétron).

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Propriedade 4: A função de onda Ψ(x, t) não é necessariamente uma função real; pode ser uma função complexa.

Isto não representa nenhum problema já que a grandeza que tem significado físico é o quadrado da função de onda, o qual é sempre um número real.

Lembremos que o quadrado da função de onda de uma partícula em cada ponto, P(x,t)=ψ*(x,t)ψ(x,t) = |ψ(x,t)|2_{, nos informa sobre a probabilidade de}

encontrar a partícula em torno desse ponto. (ψ* é o complexo conjugado de ψ_{, obtido através da substituição de i por –i onde quer que apareça).}

Específicamente, a grandeza

representa a probabilidade de encontrar a partícula representada por ψ no intervalo dx em torno da coordenada x, no instante t.

17

6-1 The Schrödinger Equation in One Dimension

233 slightly the interpretation of the wave function discussed in Chapter 5 to

accommo-date Born’s interpretation so that the probability of finding the electron in dx is real.

We take for the probability

P x, t dx

x, t

x, t dx

x, t

2

dx

6-8

where , the complex conjugate of , is obtained from by replacing i with i*

wherever it appears.

6

The complex nature of serves to emphasize the fact that we

should not ask or try to answer the question “What is waving in a matter wave?” or

inquire as to what medium supports the motion of a matter wave. The wave function

is a computational device with utility in Schrödinger’s theory of wave mechanics.

Physical significance is associated not with

itself, but with the product

2

_{, which is the probability distribution P(x, t) or, as it is often called, the}

probability density. In keeping with the analogy with classical waves and wave

func-tions, (x, t) is also sometimes referred to as the probability density amplitude, or just

the probability amplitude.

The probability of finding the electron in dx at x

₁

or in dx at x

₂

is the sum of

sepa-rate probabilities, P(x

₁

) dx P(x

₂

) dx. Since the electron must certainly be somewhere

in space, the sum of the probabilities over all possible values of x must equal 1. That is,

7

dx

1 6-9

Equation 6-9 is called the normalization condition. This condition plays an important

role in quantum mechanics, for it places a restriction on the possible solutions of the

Schrödinger equation. In particular, the wave function (x, t) must approach zero

suf-ficiently fast as x

{ so that the integral in Equation 6-9 remains finite. If it does

not, then the probability becomes unbounded. As we will see in Section 6-3, it is this

restriction together with boundary conditions imposed at finite values of x that leads

to energy quantization for bound particles.

In the chapters that follow, we are going to be concerned with solutions to the

Schrödinger equation for a wide range of real physical systems, but in what follows in

this chapter our intent is to illustrate a few of the techniques of solving the equation and

to discover the various, often surprising properties of the solutions. To this end we will

focus our attention on single-particle, one-dimensional problems, as noted earlier, and

use some potential energy functions with unrealistic physical characteristics, for

exam-ple, infinitely rigid walls, which will enable us to illustrate various properties of the

solutions without obscuring the discussion with overly complex mathematics. We will

find that many real physical problems can be approximated by these simple models.

Separation of the Time and Space Dependencies

of (

x, t )

Schrödinger’s first application of his wave equation was to problems such as the

hydrogen atom (Bohr’s work) and the simple harmonic oscillator (Planck’s work), in

which he showed that the energy quantization in those systems can be explained

natu-rally in terms of standing waves. We referred to these in Chapter 4 as stationary states,

meaning they did not change with time. Such states are also called eigenstates. For

such problems that also have potential energy functions that are independent of time,

the space and time dependence of the wave function can be separated, leading to a

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18

Capítulo 40 — Mecânica quântica I: funções de onda 283

que golpeiam o entorno desse ponto. Uma alternativa seria a probabilidade de que qualquer fóton individualmente atinja o entorno do ponto. Assim, o quadrado da magnitude do campo elétrico em cada ponto é proporcional à probabilidade de encontrar um fóton em torno desse ponto.

Da mesma maneira, o quadrado da função de onda de uma partícula em cada ponto nos informa sobre a probabilidade de encontrar a partícula em torno desse ponto. Mais precisamente, deveríamos dizer o quadrado do valor absoluto da fun-ção de onda, |C|2. Isso é necessário porque, como vimos, a função de onda é uma grandeza complexa com partes real e imaginária.

Para uma partícula que pode se mover apenas ao longo do eixo x, a grandeza |C(x, t)|2 é a probabilidade de que a partícula seja encontrada no tempo t em uma coordenada entre x e x ! dx. É mais provável a partícula ser encontrada em regiões onde |C|2 é grande, e assim por diante. Essa interpretação, elaborada pela primeira vez pelo físico alemão Max Born (Figura 40.4), exige que a função de onda C seja

normalizada. Ou seja, a integral de |C(x, t)|2dx sobre todos os valores possíveis

de x deve ser exatamente igual a 1. Em outras palavras, a probabilidade de que a partícula esteja em algum lugar é exatamente 1, ou 100%.

ATENÇÃO Interpretando |C|2 Observe que |C(x, t)|2 em si não é uma probabilidade. Por outro lado, |C(x, t)|2dx é a probabilidade de encontrarmos a partícula entre a posição

x e a posição x ! dx no tempo t. Se diminuirmos a distância dx, torna-se menos provável que a partícula será encontrada dentro desse comprimento, de modo que a probabilidade diminui. Um nome mais adequado para |C(x, t)|2 é a função de distribuição de

pro-babilidade, uma vez que descreve como a probabilidade de encontrar a partícula em

diferentes locais é distribuída no espaço. Outro nome comum para |C(x, t)|2 é densidade

de probabilidade.

Podemos utilizar a interpretação probabilística de |C|2 para chegar a uma melhor compreensão da Equação 40.18, a função de onda para uma partícula livre. Essa função descreve uma partícula que possui um momento definido p " Uk na direção do eixo x e nenhuma incerteza no momento: #p_x " 0. O princípio da incerteza de Heisenberg para a posição e o momento, equações 39.29, nos diz que #x#p_x_{$ U/2.} Se #p_x é zero, #x deve ser infinito e não temos ideia de onde, ao longo do eixo x, a partícula pode ser encontrada (vimos um resultado semelhante para os fótons na Seção 38.4). Podemos demonstrar isso pelo cálculo da função de distribuição de probabilidade |C(x, t)|2: o produto de C e seu conjugado complexo C*. Para encon-trar o complexo conjugado de um número complexo, substituímos cada i por –i. Por exemplo, o conjugado complexo de c " a ! ib, onde a e b são reais, é c* " a – ib, de modo que |c|2 " c*c " (a ! ib) (a – ib) " a2 ! b2 (lembre-se de que i2 " –1). O complexo conjugado da Equação 40.18 é

C*(x, t) " A*e-i(kx– vt) " A*e–ikxeivt

(Temos de considerar a possibilidade de que o coeficiente A possa ser, ele mesmo, um número complexo.) Dessa forma, a função de distribuição de proba-bilidades é

|C(x, t)|2 " C*(x, t) C(x, t) " (A*e–ikxeivt) (Aeikxe–ivt) " A* Ae0 " |A|2

A função de distribuição de probabilidade não depende da posição, o que sig-nifica que existe a mesma probabilidade de encontrar a partícula em qualquer

lugar ao longo do eixo x! Matematicamente, isso ocorre porque a função de onda

senoidal C(x, t) " Aei(kx– vt) " A[cos(kx % vt) ! i sen(kx % vt)] se estende por todo o caminho desde x " –∞ até x " !∞ com a mesma amplitude A. Isso também

Figura 40.4 Em 1926, o físico alemão Max Born (1882-1970)

inventou a interpretação de que |C|2 é a função de distribuição de

probabilidade de uma partícula que é descrita pela função de onda C. Ele também cunhou o termo

“mecânica quântica” (no original alemão, Quantenmechanik). Por suas contribuições, Born

compartilhou com Walther Bothe o Prêmio Nobel de física de 1954.

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Condição de normalização

19

A probabilidade de encontrar a partícula na região do espaço compreendida entre os pontos x=a e x=b é dada por:

Em particular, probabilidade de encontrar a partícula em qualquer ponto do espaço, i.e. entre x=–∞ e x=+∞ é:

Como a partícula deve estar em algum lugar, devemos ter: P(–∞, +∞, t) = 1

P (a, b, t) =

Z

b a

P (x, t)dx =

Z

b a ⇤

_{(x, t) (x, t)dx}

P ( _{1, +1, t) =} Z +₁ 1 ⇤_{(x, t) (x, t)dx}

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20

Portanto, função de onda deve verificar a condição de normalização:

Esta condição desempenha um papel importante na mecânica quântica, pois coloca uma restrição sobre as possíveis soluções da equação de Schrödinger.

Em particular, a função de onda ψ(x, t) deve se aproximar de zero suficientemente rápido quando x →±∞ {de modo que a integral acima permaneça finita. Se não, a probabilidade torna-se ilimitada.

Veremos mais adiante que essa restrição, juntamente com as condições de contorno impostas na equação de Schrödinger, levam à quantificação de energia para partículas ligadas.

Z

+₁ 1

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21

Na sequência, iremos concentrar a nossa atenção em problemas unidimensionais envolvendo apenas uma partícula.

Usaremos algumas funções de energia potencial pouco realistas, por exemplo, paredes infinitamente rígidas, o que nos permitirá ilustrar várias propriedades das soluções sem obscurecer a discussão com matemática excessivamente complexa.

Vamos descobrir que muitos problemas físicos reais podem ser aproximados por esses modelos simples.

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22

A equação de Schrödinger pode ser simplificada enormemente em situações onde a energia potencial depende das coordenadas espaciais, mas não depende explicitamente do tempo.

Nesses casos podemos usar o método de separação de variáveis e escrever a função de onda como um produto de duas funções, uma de x e uma de t: Substituindo a expressão anterior na equação de Schrödinger, temos:

onde as derivadas são agora derivadas totais e não derivadas parciais.

A equação de Schrödinger independente do tempo

greatly simplified form of the Schrödinger equation.8 The separation is accomplished by first assuming that (x, t) can be written as a product of two functions, one of x and one of t, as

x, t x t 6-10

If Equation 6-10 turns out to be incorrect, we will find that out soon enough, but if the potential function is not an explicit function of time, that is, if the potential is given by V(x), our assumption turns out to be valid. That this is true can be seen as follows:

Substituting (x, t) from Equation 6-10 into the general, time-dependent Schrödinger equation (Equation 6-6) yields

2 2m 2 _x _t x2 V x x t i x t t 6-11 which is 2 2m t d2 x dx2 V x x t i x d t dt 6-12

where the derivatives are now ordinary rather than partial ones. Dividing Equation 6-12 by in the assumed product form gives

2 2m 1 x d2 x dx2 V x i 1 t d t dt 6-13

Notice that each side of Equation 6-13 is a function of only one of the independent variables x and t. This means that, for example, changes in t cannot affect the value of the left side of Equation 6-13, and changes in x cannot affect the right side. Thus, both sides of the equation must be equal to the same constant C, called the separation

con-stant, and we see that the assumption of Equation 6-10 is valid—the variables have

been separated. In this way we have replaced a partial differential equation containing two independent variables, Equation 6-6, with two ordinary differential equations each a function of only one of the independent variables:

2 2m 1 x d2 x dx2 V x C 6-14 i 1 t d t dt C 6-15

Let us solve Equation 6-15 first. The reason for doing so is twofold: (1) Equation 6-15 does not contain the potential energy V(x); consequently, the time-dependent part

t of all solutions (x, t) to the Schrödinger equation will have the same form

when the potential is not an explicit function of time, so we only have to do this once. (2) The separation constant C has particular significance that we want to discover before we tackle Equation 6-14. Writing Equation 6-15 as

d t t C i dt iC dt 6-16

the general solution of Equation 6-16 is

t e iCt 6-17a

TIPLER_06_229-276hr.indd 234 8/22/11 11:57 AM

x, t x t 6-10

2 2m 1 x d2 x dx2 V x i 1 t d t dt 6-13

t e iCt 6-17a

TIPLER_06_229-276hr.indd 234 8/22/11 11:57 AM

x, t x t 6-10

2 2m 1 x d2 x dx2 V x i 1 t d t dt 6-13

t e iCt 6-17a

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23

Dividindo ambos lados da equação por 𝜓(x)𝜙(t), temos:

Observe que cada lado da equação acima é uma função de apenas uma das variáveis independentes x e t.

Isso significa que, por exemplo, mudanças em t não podem afetar o valor do lado esquerdo da equação e as mudanças em x não podem afetar o lado direito.

Assim, ambos os lados da equação devem ser iguais à mesma constante C, chamada de constante de separação.

234 greatly simplified form of the Schrödinger equation.

8

The separation is accomplished

by first assuming that (x, t) can be written as a product of two functions, one of x

and one of t, as

x, t

x

t

6-10

If Equation 6-10 turns out to be incorrect, we will find that out soon enough, but if

the potential function is not an explicit function of time, that is, if the potential is

given by V(x), our assumption turns out to be valid. That this is true can be seen as

follows:

Substituting

(x, t) from Equation 6-10 into the general, time-dependent

Schrödinger equation (Equation 6-6) yields

2

2m

2

_x

_t

x

2

V x

x

t

i

x

t

6-11

which is

2

2m

t

d

2

x

dx

2

V x

x

t

i

x

d

t

dt

6-12

where the derivatives are now ordinary rather than partial ones. Dividing Equation 6-12

by in the assumed product form

gives

2

2m

1 x

d

2

x

dx

2

V x

i

1 t

d

t

dt

6-13

Notice that each side of Equation 6-13 is a function of only one of the independent

variables x and t. This means that, for example, changes in t cannot affect the value of

the left side of Equation 6-13, and changes in x cannot affect the right side. Thus, both

sides of the equation must be equal to the same constant C, called the separation

con-stant, and we see that the assumption of Equation 6-10 is valid—the variables have

been separated. In this way we have replaced a partial differential equation containing

two independent variables, Equation 6-6, with two ordinary differential equations

each a function of only one of the independent variables:

2

2m

1 x

d

2

x

dx

2

V x

C

6-14

i

1 t

d

t

dt

C

6-15

Let us solve Equation 6-15 first. The reason for doing so is twofold: (1) Equation 6-15

does not contain the potential energy V(x); consequently, the time-dependent part

t of all solutions (x, t) to the Schrödinger equation will have the same form

when the potential is not an explicit function of time, so we only have to do this once.

(2) The separation constant C has particular significance that we want to discover

before we tackle Equation 6-14. Writing Equation 6-15 as

d

t

C

i

dt

iC

dt

6-16

the general solution of Equation 6-16 is

t

e

iCt

6-17a

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24

Desta forma, nós substituímos uma equação diferencial parcial contendo duas variáveis independentes, por duas equações diferenciais ordinárias, uma para cada uma das variáveis independentes:

x, t x t 6-10

2 2m 1 x d2 x dx2 V x i 1 t d t dt 6-13

t e iCt 6-17a

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25

Vamos resolver primeiramente a equação para função 𝜙(t).

O motivo é duplo:

- a equação para 𝜙(t) não depende da energia potencial V(x); portanto, a função 𝜙(t) deve ter a mesma forma para todas as soluções ψ(x,t) da equação de Schrödinger, desde que o potencial não dependa o tempo. - Outro lado, a constante de separação C tem um significado particular que

queremos descobrir antes de resolver a equação para 𝜓(x). Reescrevendo a equação anterior, temos:

x, t x t 6-10

2 2m 1 x d2 x dx2 V x i 1 t d t dt 6-13

t e iCt 6-17a

TIPLER_06_229-276hr.indd 234 8/22/11 11:57 AM

x, t x t 6-10

2 2m 1 x d2 x dx2 V x i 1 t d t dt 6-13

t e iCt 6-17a