Introduction to the Method of Quantum Mechanics

量子力學的處理方法

薛丁格理論 ( The Schrodinger Theory )

•Quantum Mechanics ( 1925 )

Schrodinger微分方程 ( differential equation )

Heisenberg矩陣 ( matrix )

•如何在平面座標上表示出一個複數?

ei= cosi sin( Euler’s identity )

Aei= A ( cosi sin)

complex conjugate :

( Ae+i)* = Ae-i

•Ae+i( kx –t ) = A [ cos( kx –t ) + i sin( kx –t ) ]

free particle wave function, ( x,t )

A

Ae+i=A ( cos+ i sin)

( A cos

(Ae+i)* = Ae-i=A ( cos- i sin)

( A sin

複數平面

實數軸

虛數軸

•Ae+i( kx –t ) = A [ cos( kx –t ) + i sin( kx –t ) ]

free particle wave function, ( x,t )

( 不同於“物理波”, 不必一定要是實數 )

= A , 2 = A2 所有位置出現機率相同

此複數形式波函數有“實數波”及“虛數波”兩個部分 , 此兩部份均具有

明確及相同的波長及頻率 :

= 2/ k

= / 2

因此所描述的粒子應該具有確定的動量及能量 :

p = h /

E = h

•對此複數形式波函數 , 還有另一個求算其動量及動能的方法

•/x = ik A ei(kx-t)

( / x : partial derivative , 偏微分, with respect to x with other

variables fixed )

-ih / x = hk A ei(kx-t) = (h/2) (2/)

-ih /x = p

-ih / x p ( -ih / x : an operator , 運算子 )

•/t = - iA ei(kx-t)

( / t : partial derivative , 偏微分 , with respect to t with other

variables fixed )

ih / t = hA ei(kx-t) = ( h/2) 2

ih /t = E

ih / t E ( ih / t : an operator , 運算子 )

Time-dependent Schrodinger Equation

•Schrodinger equation A first principle ! ( 用猜的啦 , 猜對了嘛! )

As Newton’s Law of motion, can not be mathematically derived.

•total energy of a particle

E = K.E. + P.E. = ½ mv2 + Ep = p2/2m + Ep

利用 ( p - ih / x ) 及 ( E ih / t ) ( 用猜的啦 )

E= ( p2/2m ) + Ep

ih / t = —— ( -ih —— ) ( -ih —— ) + Ep

ih / t = –—— —— + Ep

–———— + Ep= ih ——

— one-dimensional time-dependent Schrodinger equ.

12m

x

x

2x2

h2

2m

h2

2m2x2

t

Erwin Schrodinger( 1887 –1961 )

自由粒子 ( For a Free Particle )

•Ep( x ) = constant no force on the particle

Ep( x ) = constant F = 0 on a particle

in wave mechanics in particle mechanics

consider Ep consider F

•選擇 Ep = 0

–———— = ih ——

= A ei(kx-t) is the solution

h2

2m2x2

t

= A ei( kx –t )

/ x = / x [ A ei(kx-t) ] = ik A ei(kx-t)

2/ x2 = ik / x [ A ei(kx-t) ] = - k2 A ei(kx-t) = - k2

/ t = / t [ A ei(kx-t) ] = -iA ei(kx-t) = - i

- —— —— = ih / t

———= h

——— = h

= A ei( kx –t ) a particle with definite p = hk and E = h

E = h= p2 / 2m = ( h2k2 ) / 2m

h2 22m x2

h2 k2

2m

h2 k2

2m

•= A ei( kx –t ) 這樣一個波函數 , 它所代表的粒子在各點位置上出現的

機率是多少 ?

= amplitude of the wave

2 ( probability density, 機率密度 ) 粒子出現機率

( x ) 2dx = 在dx 範圍內發現粒子出現的機率

If ( x,t ) = A sin( kx –t ) , 2 = A2

If ( x,t ) = Aei( kx –t ) , ( x,t ) 2 = *

( * : complex conjugate of Y )

= A* e-i(kx-wt) A ei(kx-wt)

= A* A

= A2

( independent of x and t )

期望值 ( Expectation Value )

when x 0 :

P( x ) x = 1 P(x) dx = 1

例 : P, the probability density ( 機率密度 ) of a particle’s position( at a certain time, or assume P independent of time )

P(x)

P(xi)

xi

x( or dx )

x

the probability of finding the

particle located within dx, or

x, centered around x

= P(x)dx

P( x ) x = 1 when x 0

-

x = - [ P(- ) x ] + … + xn [ P(xn) x ] + ….. + [ P() x ]

x = xi [ p(xi) x ]

x 0 , x dx

x = x P(x) dx

P( x ) = 2 = * , P(x) dx = * dx , *dx = 1

x = x * dx

= * x dx

similarly,

Ep = * Ep(x) dx

P(x)

P(xn)

xn

x( or dx )

x

-

-

-

-

-

•to find p 利用 p 的運算子 ( -i h —— )

p = * ( -ih / x ) dx ( * p dx )

x

例 : if = A ei( kx –t ) ( a free particle )

p = * ( -ih / x ) A ei(kx-t) dx

= * [ -ih ik A ei(kx-t) ] dx

= hk * dx

= hk = ( h/2) ( 2/) = h/

-

-

-

-

-

•to find E 利用 E 的運算子 ( i h —— )

E = * ( ih / t ) dx ( * E dx )

t

例 : if = A ei( kx –t ) ( a free particle )

E = * ( ih / t ) A ei(kx-t) dx

= * [ ih ( -i) A ei(kx-t) ] dx

= h * dx

= h= ( h/2) ( 2) = h

-

-

-

-

-

Short Summary ( 以下在開始時都是猜的 ) :

•can be a function involving complex numbers

•when is a function of complex numbers

(1) the probability of particle appearance * = 2

(2) the position expectation value x = * x dx

(3) ( -ih / x )= p

the momentum expectation value p = * ( -ih / x ) dx

(4) ( ih / t )= E

the energy expectation value E = * ( ih / t ) dx

(5) –———— + Ep= ih ——

以上猜測後來經過證明都猜對了 !

-

-

-

h2

2m2x2

t

Time-Independent Schrodinger Equation

•一種解薛丁格方程式的數學方法 變數分離法 ( Separation of Variables )

首先假設 ( x,t ) = ( x ) ( t )

– + Ep(x) = ih / t

( t ) • + Ep(x) ( x) ( t ) ih ( x )

( x ) ( t ) on both sides of the equation

– + Ep(x) = ih —————

– + Ep(x) = G

ih ————— = G , G : separation constant

h2

2m2x2

h2

2md2( x)

dx2d( t )

dt

h2

2m1

( x)d2( x)

dx21

( t )d( t )

dt

h2

2m1

( x)

d2( x)

dx2

1( t )

d( t )dt

•to solve ( t )

d ( t ) / dt = ( G / ih )

( t ) = k e( G / ih )t = k e( -iG /h )t = e( -iGt )/h ( choose k = 1 )

G = ? ih —( x,t ) = ih —( x ) ( t )

= ih —( x ) e( -iGt )/h

= ih ( x ) ( -iG / h ) e( -iGt )/h

= G ( x ) ( t )

= G ( x,t )

ih — E

G = E , ( t ) = e( -i E t / h ) independent of Ep( x ) ,

fixed form for all physical situations

t

t

t

t

•space-dependent part of ( x,t )

– + Ep(x) = E

— time-independent Schrodinger equation

: eigenfunction or eigenstate

E : eigenvalues

( German “eigen”means “proper”)

h2

2md2dx2

( x ) 及 d/ dx 必須具備的三個特性

( Required Properties of ( x ) and d/ dx )

(1) ( x ) 及 d/ dx 在所有位置都必須是有限值

例 : 假如 ( x ) 或 d/ dx 有無限大的值出現 :

( x ) ord/ dx

x

p = *( -ih / x ) dx

then E

physically impossible

-

(2) ( x ) 及 d/ dx 在所有位置都必須各只有一個唯一值 ( single-valued )

例 : 假如 ( x ) 或 d/ dx 在某一個位置上有不只一個的值出現 :

x = * •x •dx

p = *( -ih / x ) dx

( x ) ord/ dx

x

-

-

can not be defined

(3) ( x ) 及 d/ dx 在所有位置都必須是連續的

例 : 假如 ( x ) 或 d/ dx 在某一個位置上是不連續的 :

d/ dx infinite at xo

not allowed

or d2/ dx infinite at xo

– + Ep(x) = E

then Ep( xo ) or E, physically impossible

( x ) ord/ dx

xxo

h2

2md2dx2

因為 ( x ) 及 d/ dx 必須滿足前面這三個特性的要求

在滿足 time-indep. Schrodinger equ. 的解中

只有某些特定形式的 ( x ) 函數可以被接受

( x ) : eigenfunction, eigenstate

也只有在某些特定的 E 值被帶入薛丁格方程式中時 , 才能獲得合

理形式的 解

E : eigenvalues ( of particle total energy )

quantization of E and other physical quantities

time-independent Schrodinger equation :

–———— + Ep( x ) = E h2 d2

2m dx2

薛丁格方程式的應用範例 –無限高位能井中的粒子( Application of Schrodinger Theory –Infinite Potential Well )

Ep( x ) = , x < 0 and x > a

0 , 0 x a

•x 0 , x a

粒子不能存在2 = 0= 0

•0 x a

– + Ep(x) = E , Ep = 0

– = E —— + ———= 0

p = h/= h / (2/k) = hk , E = p2/2m = h2k2 / 2m , k2 = 2mE / h2

d2/dx2 + k2= 0 d2/dx2 = - k2

h2

2md2dx2

h2

2md2dx2

d2dx2

2mEh2

x0 a

Ep(x)

d2/dx2 = - k2 , try = ex

d/dx = ex , d2/dx2 = 2 ex = 2= -k2

= ± ik

general solution : (x) = a eikx + b e-ikx

= a ( cos kx + i sin kx ) + b ( cos kx –i sin kx )

= ( a+b ) cos kx + i ( a-b ) sin kx

= A cos kx + B sin kx

( x ) is finite and single-valued already

for ( x ) to be continuous :

( x=0 ) = 0 A = 0 ( x ) = B sin kx

( x=a ) B sin ka = 0 sin ka = 0 ( B 0 )

ka = 0, , 2, 3, ….

k = n/ a , n = 1, 2, 3, ….. ( k 0 , if k = 0= 0 )

•eigenfunction : n( x ) = B ( sin n/a ) •x

eigenvalues : En = h2k2 / 2m = ( n22 / a2 ) ( h2 / 2m )

= n2Eo , Eo = 2h2 / ( 2ma2 )

note : d/dx not continuous at x = 0 and x = a

because Ep at x = 0 and x = a

a

2

n E

3

2

1

9 Eo

4 Eo

Eo

( n = 1 : ground state )

•量子力學結果與古典力學的不同

classical theory quantum mechanics

E continuous quantized

@ T = 0 K E = 0 E = Eo 0

( 3/2 kBT : average energy

of an atom in a gas )

例 : Eo = 2h2 / ( 2ma )

(1) a marble ( m = 0.1 kg ) , a = 0.01 m

Eo = ———— = ————————— = ……

(2) an e- , a = 10 Å

Eo = ——— = ——————————— = ……

2h2 / 42 ( 6.6x10-34 )2

2ma2 8 x 0.1 x ( 0.01 )2

h2 ( 6.6 x 10-34 )2

8ma2 8 x 9.1 x 10-31 x ( 10-9 )2

例 : 1( x,t ) = 1( x ) ( t ) = B sin( x / a ) exp( - iEot / h )

* dx = 1 = * dx = 1 * 1 dx

12 dx = 1 ….. B = ( 2/a )1/2 ( * = 1 )

-

a0

a0

a0

例 : 針對 1 , 計算 x , p, E

x = 1* x 1 dx = 2/a sin2( x/a ) x dx = ….. = a/2

p = 1* ( -ih / x ) 1 dx = 1 * ( -ih 1 / x ) dx

= 1 ( -ih 1 / x ) dx

= 2/a sin( x/a )( -ih / x ) sin( x/a ) dx = ….. = 0

E = 1* ( ih / t ) 1 dx

= 1* ( -ih / t ) 1( x) exp( -iEot / h ) dx

= 1* ( -ih )( -iEo/h ) 1( x) exp( -iEot / h ) dx

= Eo 1* 1 dx = Eo ( 1* 1 dx = 1 )

a0

a0

a0

a0

a0

a0

a0a0a0a0

a0

( Q : The expectation value of x seems different from theclassical anticipation. Does this contradict classicalmechanics ? )

簡諧振盪體 ( The Harmonic Oscillator )

in wave mechanics in classical mechanics

consider Ep consider F

Ep( x ) = ½ kx2 F = - k x

one-dim. time-indep. Schrodinger equ.

– + ½ k x2 = E h2

2m

d2

dx2

equilibrium

F

F x

x

Ep(x)

必須滿足三個特性 ( 有限 , 唯一 , 連續 ) 的要求

只有在 E 等於某些特定值時, 才能解出合理的函數

En = ( n + ½ ) h ( k/m )1/2 n ( Hermite functions )

( n = 0, 1, 2, 3 , ….. )

幾個重要的觀察 :

(1) ( k/m )1/2 = En = ( n + ½ ) h= ( n + ½ ) h

(2) difference between adjacent energy levels is a constant h

consistent with Planck’s blackbody theory

(3 ) Eo = ½ h ( 0 ) , E = h

different from Planck’s blackbody theory ( i.e. E = nh, Eo = 0 )

different from energy levels in atoms or infinite potential wells

o = C exp ( - mk x2 / 2h ) : ground state

C can be decided through 2dx = 1 C = ( m/ h )1/4

(5) in classical mechanics, if E = ½ h E = ½ kxmax2 , xmax=( 2E/k )½

in classical mechanics,

the particle can not exceed xmax

but in quantum mechanics,

the particle may exceed xmax

( with low probabilities )

-

xmax- xmax

o(x)

x

( Q : The expectation value of x seems different from the classicalanticipation. Does this contradict classical mechanics ? )

有限高位能井中的粒子 ( Finite Potential Well )

without the infinite potential change at the boundaries

and d/dx should be continuous at all positions

d/dx will bediscontinuousat boundaries

d/dx will becontinuous

at boundaries

The particle does have the possibility ofovercoming the potential barrier even whenE is smaller than the energy barrier.

one-dim. time-indep. Schrodinger equ.

– + Ep( x ) = E h2

2m

d2

dx2

量子穿隧效應 ( Quantum Tunneling )

thick potentialbarrier

thin potentialbarrier

The function inside the barrier

exp[ – 2m( Uo-E ) x / h ]

( Uo : barrier height )

m large unlikely to see tunneling

x

Scanning Tunneling Microscope

e-

Gerd Binnig and Heinrich Rohrer( 1986 Nobel Laureates )