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? 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 )
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
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 )
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.
Erwin Schrodinger ( 1887 1961 )
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
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
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 )
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 -
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
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
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
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!
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
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
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 )
( 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
(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
(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
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
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
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 )
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 3 2 1 ( 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
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 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 10-31 x ( 10-9 ) 2
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 - 0 0 a 2 1 dx = 1.. B = ( 2/a ) 1/2 ( * = 1 ) 0
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. 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 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
( Q : The expectation value of x seems different from the classical anticipation. 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 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
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
Scanning Tunneling Microscope Gerd Binnig and Heinrich Rohrer ( 1986 Nobel Laureates ) e -