第四章牛頓定律

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1 Introduction to the Method of Quantum Mechanics 量子力學的處理方法

2 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? e iq = cosq i sinq ( Euler s identity ) 虛數軸 Ae +iq = A ( cosq + i sinq ) Ae iq = A ( cosq i sinq ) complex conjugate : ( A sinq ) A q - q 實數軸 ( A cosq ) ( Ae +iq )* = Ae -iq 複數平面 (Ae +iq )* = Ae -iq = A ( cosq - i sinq ) Ae +i( kx wt ) = A [ cos( kx wt ) + i sin( kx wt ) ] free particle wave function, Y( x,t )

3 Ae +i( kx wt ) = A [ cos( kx wt ) + i sin( kx wt ) ] free particle wave function, Y( x,t ) ( Y 不同於物理波, 不必一定要是實數 ) Y = A, Y 2 = A 2 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 : l = 2p / k u = w / 2p 因此所描述的粒子應該具有確定的動量及能量 : p = h / l E = hu 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

4 Y/x = ik A e i(kx-wt) ( / x : partial derivative, 偏微分, with respect to x with other variables fixed ) -ih Y / x = hk A e i(kx-wt) = (h/2p) (2p/l) Y -ih Y /x = p Y -ih / x p ( -ih / x : an operator, 運算子 ) Y/t = - iw A e i(kx-wt) ( / t : partial derivative, 偏微分, with respect to t with other variables fixed ) ih Y / t = hw A e i(kx-wt) = ( h/2p ) 2p u Y ih Y /t = E Y ih / t E ( ih / t : an operator, 運算子 )

5 Time-dependent Schrodinger Equation total energy of a particle E = K.E. + P.E. = ½ mv 2 + E p = p 2 /2m + E p 利用 ( p - ih / x ) 及 ( E ih / t ) ( 用猜的啦 ) EY = ( p 2 /2m ) Y + E p Y 1 ih Y / t = ( -ih ) ( -ih ) Y + E p Y 2m x x h 2 ih Y / t = 2 Y + E p Y 2m x 2 h 2 2 Y Y + E p Y = ih 2m x 2 t one-dimensional time-dependent Schrodinger equ. Schrodinger equation A first principle! ( 用猜的啦, 猜對了嘛! ) As Newton s Law of motion, can not be mathematically derived.

6 Erwin Schrodinger ( )

7 自由粒子 ( For a Free Particle ) E p ( x ) = constant no force on the particle E p ( x ) = constant F = 0 on a particle in wave mechanics in particle mechanics consider E p consider F 選擇 E p = 0 2 Y Y = ih 2m x 2 t h 2 Y = A e i(kx-wt) is the solution

8 Y = A ei( kx wt ) Y / x = / x [ A e i(kx-wt) ] = ik A e i(kx-wt) 2 Y / x 2 = ik / x [ A e i(kx-wt) ] = - k 2 A e i(kx-wt) = - k 2 Y Y / t = / t [ A e i(kx-wt) ] = -iw A e i(kx-wt) = - iw Y - h 2 2 Y = ih Y / t 2m x 2 h 2 k 2 Y = hw Y 2m h 2 k 2 = h w 2m Y = A e i( kx wt ) a particle with definite p = hk and E = hw E = hw = p 2 / 2m = ( h 2 k 2 ) / 2m

9 Y = A e i( kx wt ) 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? Y = amplitude of the wave Y 2 ( probability density, 機率密度 ) 粒子出現機率 Y( x ) 2 dx = 在 dx 範圍內發現粒子出現的機率 If Y( x,t ) = A sin( kx wt ), Y 2 = A 2 If Y( x,t ) = Ae i( kx wt ), Y( x,t ) 2 = Y* Y ( Y* : complex conjugate of Y ) = A* e -i(kx-wt) A e i(kx-wt) = A* A = A 2 ( independent of x and t )

10 期望值 ( Expectation Value ) 例 : P, the probability density ( 機率密度 ) of a particle s position ( at a certain time, or assume P independent of time ) the probability of finding the particle located within dx, or Dx, centered around x = P(x)dx P(x i ) P(x) x i P( x ) Dx = 1 when Dx 0 Dx ( or dx ) x when Dx 0 : P( x ) Dx = 1 P(x) dx = 1 -

11 x = - [ P(- ) Dx ] + + x n [ P(x n ) Dx ] [ P( ) Dx ] x = x i [ p(x i ) Dx ] Dx 0, Dx dx x = x P(x) dx - P( x ) = Y 2 = Y* Y, P(x) dx = Y* Y dx, Y*Y dx = 1 - x = - x Y* Y dx = Y* x Y dx - similarly, E p = - Y* E p (x) Y dx P(x) x n Dx ( or dx ) P(x n ) x

12 to find p 利用 p 的運算子 ( -i h ) x p = Y* ( -ih / x ) Y dx ( Y* p Y dx ) - - 例 : if Y = A e i( kx wt ) ( a free particle ) p = - Y* ( -ih / x ) A e i(kx-wt) dx = Y* [ -ih ik A e i(kx-wt) ] dx - = hk Y* Y dx - = hk = ( h/2p ) ( 2p/l ) = h/l

13 to find E 利用 E 的運算子 ( i h ) t E = Y* ( ih / t ) Y dx ( Y* E Y dx ) - - 例 : if Y = A e i( kx wt ) ( a free particle ) E = - Y* ( ih / t ) A e i(kx-wt) dx = Y* [ ih ( -iw ) A e i(kx-wt) ] dx - = hw Y* Y dx - = hw = ( h/2p ) ( 2pu ) = hu

14 Short Summary ( 以下在開始時都是猜的 ) : Y can be a function involving complex numbers when Y is a function of complex numbers (1) the probability of particle appearance Y* Y = Y 2 (2) the position expectation value x = Y* x Y dx - (3) ( -ih / x )Y = py the momentum expectation value p = Y* ( -ih / x ) Y dx - (4) ( ih / t )Y = EY the energy expectation value E = - Y* ( ih / t ) Y dx (5) h 2 2 Y Y + E p Y = ih 2m x 2 t 以上猜測後來經過證明都猜對了!

15 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 Y( x,t ) = ( x ) ( t ) h 2 2 Y + E p (x) Y = ih Y / t 2m x 2 d 2 ( x) d( t ) - ( t ) + E p (x) ( x) ( t ) = ih ( x ) 2m dx 2 dt h 2 ( x ) ( t ) on both sides of the equation 1 d 2 ( x) 1 + E p (x) = ih d( t ) 2m ( x) dx 2 ( t ) dt h 2 h 2 1 d 2 ( x) + E p (x) = G 2m ( x) dx 2 1 ih d( t ) = G, G : separation constant ( t ) dt

16 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 Y( x,t ) = ih ( x ) ( t ) t t ih E t = ih ( x ) e t ( -igt )/h = ih ( x ) ( -ig / h ) e ( -igt )/h = G ( x ) ( t ) = G Y( x,t ) G = E, ( t ) = e ( -i E t / h ) independent of E p ( x ), fixed form for all physical situations

17 space-dependent part of Y( x,t ) h 2 d 2 + E p (x) = E 2m dx 2 time-independent Schrodinger equation : eigenfunction or eigenstate E : eigenvalues ( German eigen means proper )

18 ( 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 : p = - Y*( -ih / x ) Y dx then E ( x ) or d / dx physically impossible x

19 (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 ) or d / dx x x = Y* x Y dx - p = Y*( -ih / x ) Y dx - can not be defined

20 (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 x o not allowed ( x ) or d / dx or d 2 / dx infinite at x o h 2 d 2 + E p (x) = E 2m dx 2 x o x then E p ( x o ) or E, physically impossible

21 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 : + E p ( x ) = E h 2 d 2 2m dx 2

22 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 ) E p ( x ) =, x < 0 and x > a 0, 0 x a x 0, x a 粒子不能存在 2 = 0 = 0 E p (x) 0 x a h 2 d 2 + E p (x) = E, E p = 0 2m dx 2 h 2 d 2 d = E 2 2mE + = 0 2m dx 2 dx 2 h 2 x 0 a p = h/l = h / (2p/k) = hk, E = p 2 /2m = h 2 k 2 / 2m, k 2 = 2mE / h 2 d 2 /dx 2 + k 2 = 0 d 2 /dx 2 = - k 2

23 d 2 /dx 2 = - k 2, try = e ax d/dx = a e ax, d 2 /dx 2 = a 2 e ax = a 2 = -k 2 a = ± ik general solution : (x) = a e ikx + 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 : (1) ( x=0 ) = 0 A = 0 ( x ) = B sin kx (2) ( x=a ) B sin ka = 0 sin ka = 0 ( B 0 ) ka = 0, p, 2p, 3p,. k = np / a, n = 1, 2, 3,.. ( k 0, if k = 0 = 0 )

24 eigenfunction : n ( x ) = B ( sin np/a ) x eigenvalues : E n = h 2 k 2 / 2m = ( n 2 p 2 / a 2 ) ( h 2 / 2m ) = n 2 E o, E o = p 2 h 2 / ( 2ma 2 ) n E Y Y 2 a ( n = 1 : ground state ) 9 E o 4 E o E o note : d/dx not continuous at x = 0 and x = a because E p at x = 0 and x = a

25 量子力學結果與古典力學的不同 classical theory quantum mechanics E continuous T = 0 K E = 0 E = Eo 0 ( 3 / 2 k B T : average energy of an atom in a gas ) 例 : E o = p 2 h 2 / ( 2ma ) (1) a marble ( m = 0.1 kg ), a = 0.01 m p E o = 2 h 2 / 4p 2 ( 6.6x10 = -34 ) 2 = 2ma 2 8 x 0.1 x ( 0.01 ) 2 (2) an e -, a = 10 Å h E o = 2 ( 6.6 x 10 = -34 ) 2 = 8ma 2 8 x 9.1 x x ( 10-9 ) 2

26 例 : Y 1 ( x,t ) = 1 ( x ) ( t ) = B sin( px / a ) exp( - ie o t / h ) a a Y* Y dx = 1 = Y* Y dx = 1 * 1 dx a 2 1 dx = 1.. B = ( 2/a ) 1/2 ( * = 1 ) 0

27 例 : 針對 Y 1, 計算 x, p, E a a x = Y 1 * x Y 1 dx = 2/a sin 2 ( px/a ) x dx =.. = a/2 0 0 a a p = Y 1 * ( -ih / x ) Y 1 dx = 1 * ( -ih 1 / x ) dx 0 0 a = 1 ( -ih 1 / x ) dx 0 a = 2/a sin( px/a )( -ih / x ) sin( px/a ) dx =.. = 0 0 a E = Y 1 * ( ih / t ) Y 1 dx 0 a = Y 1 * ( -ih / t ) 1 ( x) exp( -ie o t / h ) dx 0 a = Y 1 * ( -ih )( -ie o /h ) 1 ( x) exp( -ie o t / h ) dx 0 a a = E o Y 1 * Y 1 dx = E o ( Y 1 * Y 1 dx = 1 ) 0 0 ( Q : The expectation value of x seems different from the classical anticipation. Does this contradict classical mechanics? )

簡諧振盪體 ( The Harmonic Oscillator ) in wave mechanics in classical mechanics consider E p consider F F E p ( x ) = ½ kx 2 F = - k x F equilibrium x E p (x) one-dim. time-indep.

28 簡諧振盪體 ( The Harmonic Oscillator ) in wave mechanics in classical mechanics consider E p consider F F E p ( x ) = ½ kx 2 F = - k x F equilibrium x E p (x) one-dim. time-indep. Schrodinger equ. h 2 d 2 + ½ k x 2 = E 2m dx 2 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 E n = ( 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 = w E n = ( n + ½ ) hw = ( n + ½ ) hu (2) difference between adjacent energy levels is a constant hu consistent with Planck s blackbody theory (3 ) E o = ½ hu ( 0 ), DE = hu

29 幾個重要的觀察 : (1) ( k/m ) 1/2 = w E n = ( n + ½ ) hw = ( n + ½ ) hu (2) difference between adjacent energy levels is a constant hu consistent with Planck s blackbody theory (3 ) E o = ½ hu ( 0 ), DE = hu different from Planck s blackbody theory ( i.e. E = nhu, E o = 0 ) different from energy levels in atoms or infinite potential wells (4) o = C exp ( - mk x 2 / 2h ) : ground state C can be decided through 2 dx = 1 C = ( mw / ph ) - 1/4 (5) in classical mechanics, if E = ½ hu E = ½ kx 2 max, x max =( 2E/k ) ½ in classical mechanics, the particle can not exceed x max but in quantum mechanics, o (x) the particle may exceed x max ( with low probabilities ) - x max x max x

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

at boundaries The particle does have the possibility of overcoming the potential barrier even when E

31 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 be discontinuous at boundaries d/dx will be continuous at boundaries The particle does have the possibility of overcoming the potential barrier even when E is smaller than the energy barrier. one-dim. time-indep. Schrodinger equ. h 2 d 2 + E p ( x ) = E 2m dx 2

32 量子穿隧效應 ( Quantum Tunneling ) thick potential barrier The Y function inside the barrier x exp[ 2m( U o -E ) x / h ] ( U o : barrier height ) m large unlikely to see tunneling thin potential barrier

33 Scanning Tunneling Microscope Gerd Binnig and Heinrich Rohrer ( 1986 Nobel Laureates ) e -

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第四章 牛頓定律

第四章牛頓定律