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Aplikasi Metode Beda Hingga Pada Persamaan Schrödinger Menggunakan Matlab

Lampiran 1. Kode Pemrograman Visualisasi gelombang pada partikel dengan potensial halang

  clc L=Str2num(get(handles.edit1, 'String' )); Ni=Str2num(get(handles.edit2, 'String' )); ni=Str2num(get(handles.edit6, 'String' )); m=Str2num(get(handles.edit5, 'String' )); ii=Str2num(get(handles.edit9, 'String' )); N=Ni; h=L/(N+1); set(handles.edit3, 'String' ,h); hplank=6.628*10^-34; omega=8.45*10^20; Vh=0.5*m*omega^2*h^2; Eni=5.6*10^-13*(ni+0.5); set(handles.edit4, 'string' ,Vh); set(handles.edit7, 'String' ,Eni); k2=(2*m*(Vh-Eni)/hplank^2)^0.5; k1=(2*m*(Eni)/hplank^2)^0.5; Ai=2; x1=linspace(-L,0,200); xxx=linspace(0,L,(N+2)); x3=linspace(L,2*L,200); R=(sin(k2*L))/(sin(k2*L)^2+(4*k1^2*k2^2/(k1^2-k2^2)^2)); Bi=(R*Ai^2)^0.5; Psi1=Ai*exp(i*k1*x1)+Bi*exp(-i*k1*x1); alpha=Ai*exp(i*k1*0)+Bi*exp(-i*k1*0);

  if Eni<Vh

  QQi=Ai+Bi; Psi2i=QQi*exp(-k2*xxx); beta=QQi*exp(-k2*L); Fi=beta/(exp(i*k1*2*L)); Psi3=Fi*exp(i*k1*x3);

  % Psi Analatik

  n=N;

  %Membuat Matriks A Berdiagonal Sesuai Penyelesaian Finit Beda Hingga for ii=1:n

  A(ii,ii)=(2+h^2*((k2)^2));

  end for ii=1:n-1

  A(ii,ii+1)=-1;

  end for ii=2:n

  A(ii,ii-1)=-1;

  end % Penyusunan Vektor berdiri b

  b(1,1)=alpha;

  for ii=2:n-1

  b(ii,1)=0;

  end

  b(n,1)=beta;

  %Kalibrasi b kedalam A for ii=1:n

  A(ii,n+1)=b(ii,1);

  end for j=1:(n-1) % mulai proses pivot if (A(j,j)==0) for p=1:n+1 u=A(j,p); v=A(j+1,p); A(j+1,p)=u; A(j,p)=v;

  end end %akhir proses pivot

  jj=j+1;

  for ii=jj:n

  m=A(ii,j)/A(j,j);

  for k=1:(n+1)

  A(ii,k)=A(ii,k)-(m*A(j,k));

  end end end %Proses Substitusi mundur

  x(n,1)=A(n,n+1)/A(n,n);

  for ii=n-1:-1:1

  S=0;

  for j=n:-1:ii+1

  S=S+A(ii,j)*x(j,1);

  end

  x(ii,1)=(A(ii,n+1)-S)/A(ii,ii);

  end %Menampilkan Vektor ww

  ww=zeros(1,n+2); ww(1,1)=real(alpha); ww(1,N+2)=real(beta);

  for uu=2:N+1

  ww(:,uu)=real(x(uu-1,:))';

  end

  P=zeros(1,2*N); Pi=zeros(1,2*N); NN=N+2;

  for oo=1:2:2*NN;

  P(:,oo)=(xxx((oo+1)/2));

  end for ooo=2:2:2*NN;

  P(:,ooo)=(xxx(ooo/2));

  end for oooo=1:4:2*NN;

  Pi(:,oooo)=10;

  end for ooooo=2:4:2*NN;

  Pi(:,ooooo)=-10;

  end for oooooo=3:4:2*NN;

  Pi(:,oooooo)=-10;

  end for ooooooo=4:4:2*NN;

  Pi(:,ooooooo)=10;

  end

  format long e Transposex=xxx'; TranposePsi2=ww'; Tampil=[Transposex TranposePsi2 real(Psi2i)'] plot(P,Pi, 'g' ,x1,real(Psi1),xxx,ww, 'r' ,x3,real(Psi3)); axis([-L 2*L -8 8])

  else

  C=(alpha/2)+((i*k1/(2*k2))*(Ai-Bi)); D=(alpha/2)-((i*k1/(2*k2))*(Ai-Bi));

  Psi2=C*exp(k2*xxx)+D*exp(-k2*xxx); real(C) real(D) yyy=C*exp(k2*(0+h))+D*exp(-k2*(0+h)) beta=C*exp(k2*L)+D*exp(-k2*L); Fi=beta/(exp(i*k1*L)); Psi3=Fi*exp(i*k1*x3);

  x(n,1)=A(n,n+1)/A(n,n);

  jj=j+1;

  for ii=jj:n

  m=A(ii,j)/A(j,j);

  for k=1:(n+1)

  A(ii,k)=A(ii,k)-(m*A(j,k));

  end end end %Proses Substitusi mundur

  for ii=n-1:-1:1

  u=A(j,p); v=A(j+1,p); A(j+1,p)=u; A(j,p)=v;

  S=0;

  for j=n:-1:ii+1

  S=S+A(ii,j)*x(j,1);

  end

  x(ii,1)=(A(ii,n+1)-S)/A(ii,ii);

  end %Menampilkan Vektor ww

  end end %akhir proses pivot

  end for j=1:(n-1) % mulai proses pivot if (A(j,j)==0) for p=1:n+1

  % Psi Analatik

  A(ii,ii-1)=-1;

  n=N;

  %Membuat Matriks A Berdiagonal Sesuai Penyelesaian Finit Beda Hingga for ii=1:n

  A(ii,ii)=(2+h^2*((k2)^2));

  end for ii=1:n-1

  A(ii,ii+1)=-1;

  end for ii=2:n

  end % Penyusunan Vektor berdiri b

  A(ii,n+1)=b(ii,1);

  b(1,1)=alpha;

  for ii=2:n-1

  b(ii,1)=0;

  end

  b(n,1)=beta;

  %Kalibrasi b kedalam A for ii=1:n

  ww=zeros(1,n+2); ww(1,1)=real(alpha); ww(1,N+2)=real(beta); for uu=2:N+1

  ww(:,uu)=real(x(uu-1,:))';

  end

  P=zeros(1,2*N); Pi=zeros(1,2*N); NN=N+2;

  for oo=1:2:2*NN;

  P(:,oo)=(xxx((oo+1)/2));

  end for ooo=2:2:2*NN;

  P(:,ooo)=(xxx(ooo/2));

  end for oooo=1:4:2*NN;

  Pi(:,oooo)=10;

  end for ooooo=2:4:2*NN;

  Pi(:,ooooo)=-10;

  end for oooooo=3:4:2*NN;

  Pi(:,oooooo)=-10;

  end for ooooooo=4:4:2*NN;

  Pi(:,ooooooo)=10;

  end

  format long e Transposex=xxx'; TranposePsi2=ww'; Tampil=[Transposex TranposePsi2 real(Psi2)'] plot(P,Pi, 'g' ,x1,real(Psi1),xxx,ww, 'r' ,x3,real(Psi3)); axis([-L 2*L -8 8])

  end case

  2 clc L=Str2num(get(handles.edit1, 'String' )); Ni=Str2num(get(handles.edit2, 'String' )); ni=Str2num(get(handles.edit6, 'String' )); m=Str2num(get(handles.edit5, 'String' )); ii=Str2num(get(handles.edit9, 'String' )); N=Ni; h=L/(N+1); set(handles.edit3, 'String' ,h); hplank=6.628*10^-34; omega=8.45*10^20; Vh=0.5*m*omega^2*h^2; Eni=5.6*10^-13*(ni+0.5); set(handles.edit4, 'string' ,Vh); set(handles.edit7, 'String' ,Eni); k2=(2*m*(Vh-Eni)/hplank^2)^0.5; k1=(2*m*(Eni)/hplank^2)^0.5; Ai=2; x1=linspace(-L,0,200); xxx=linspace(0,L,(N+2)); x3=linspace(L,2*L,200); R=(sin(k2*L))/(sin(k2*L)^2+(4*k1^2*k2^2/(k1^2-k2^2)^2)); Bi=(R*Ai^2)^0.5; Psi1=Ai*exp(i*k1*x1)+Bi*exp(-i*k1*x1); alpha=Ai*exp(i*k1*0)+Bi*exp(-i*k1*0);

  if Eni<Vh

  QQi=Ai+Bi; Psi2i=QQi*exp(-k2*xxx); beta=QQi*exp(-k2*L);

  Fi=beta/(exp(i*k1*2*L)); Psi3=Fi*exp(i*k1*x3);

  for j=n:-1:ii+1

  m=A(ii,j)/A(j,j);

  for k=1:(n+1)

  A(ii,k)=A(ii,k)-(m*A(j,k));

  end end end %Proses Substitusi mundur

  x(n,1)=A(n,n+1)/A(n,n);

  for ii=n-1:-1:1

  S=0;

  S=S+A(ii,j)*x(j,1);

  jj=j+1;

  end

  x(ii,1)=(A(ii,n+1)-S)/A(ii,ii);

  end %Menampilkan Vektor ww

  ww=zeros(1,n+2); ww(1,1)=real(alpha); ww(1,N+2)=real(beta);

  for uu=2:N+1

  ww(:,uu)=real(x(uu-1,:))';

  end

  for ii=jj:n

  end end %akhir proses pivot

  % Psi Analatik

  end % Penyusunan Vektor berdiri b

  n=N;

  %Membuat Matriks A Berdiagonal Sesuai Penyelesaian Finit Beda Hingga for ii=1:n

  A(ii,ii)=(2+h^2*((k2)^2));

  end for ii=1:n-1

  A(ii,ii+1)=-1;

  end for ii=2:n

  A(ii,ii-1)=-1;

  b(1,1)=alpha;

  u=A(j,p); v=A(j+1,p); A(j+1,p)=u; A(j,p)=v;

  for ii=2:n-1

  b(ii,1)=0;

  end

  b(n,1)=beta;

  %Kalibrasi b kedalam A for ii=1:n

  A(ii,n+1)=b(ii,1);

  end for j=1:(n-1) % mulai proses pivot if (A(j,j)==0) for p=1:n+1

  P=zeros(1,2*N); Pi=zeros(1,2*N);

  NN=N+2;

  for oo=1:2:2*NN;

  P(:,oo)=(xxx((oo+1)/2));

  end for ooo=2:2:2*NN;

  P(:,ooo)=(xxx(ooo/2));

  end for oooo=1:4:2*NN;

  Pi(:,oooo)=10;

  end for ooooo=2:4:2*NN;

  Pi(:,ooooo)=-10;

  end for oooooo=3:4:2*NN;

  Pi(:,oooooo)=-10;

  end for ooooooo=4:4:2*NN;

  Pi(:,ooooooo)=10;

  end

  format long e Transposex=xxx'; TranposePsi2=ww'; plot(P,Pi, 'y' ,x1,real(Psi1),xxx,real(Psi2i),x3,real(Psi3)); axis([-L 2*L -8 8])

  else

  C=(alpha/2)+((i*k1/(2*k2))*(Ai-Bi)); D=(alpha/2)-((i*k1/(2*k2))*(Ai-Bi)); Psi2=C*exp(k2*xxx)+D*exp(-k2*xxx); beta=C*exp(k2*L)+D*exp(-k2*L); Fi=beta/(exp(i*k1*L)); Psi3=Fi*exp(i*k1*x3);

  % Psi Analatik

  n=N;

  %Membuat Matriks A Berdiagonal Sesuai Penyelesaian Finit Beda Hingga for ii=1:n

  A(ii,ii)=(2+h^2*((k2)^2));

  end for ii=1:n-1

  A(ii,ii+1)=-1;

  end for ii=2:n

  A(ii,ii-1)=-1;

  end % Penyusunan Vektor berdiri b

  b(1,1)=alpha;

  for ii=2:n-1

  b(ii,1)=0;

  end

  b(n,1)=beta;

  %Kalibrasi b kedalam A for ii=1:n

  A(ii,n+1)=b(ii,1);

  end for j=1:(n-1) % mulai proses pivot if (A(j,j)==0) for p=1:n+1

  u=A(j,p); v=A(j+1,p); A(j+1,p)=u; A(j,p)=v; end end %akhir proses pivot

  jj=j+1;

  for ii=jj:n

  m=A(ii,j)/A(j,j);

  for k=1:(n+1)

  A(ii,k)=A(ii,k)-(m*A(j,k));

  end end end %Proses Substitusi mundur

  x(n,1)=A(n,n+1)/A(n,n);

  for ii=n-1:-1:1

  S=0;

  for j=n:-1:ii+1

  S=S+A(ii,j)*x(j,1);

  end

  x(ii,1)=(A(ii,n+1)-S)/A(ii,ii);

  end %Menampilkan Vektor ww

  ww=zeros(1,n+2); ww(1,1)=real(alpha); ww(1,N+2)=real(beta);

  for uu=2:N+1

  ww(:,uu)=real(x(uu-1,:))';

  end

  P=zeros(1,2*N); Pi=zeros(1,2*N); NN=N+2;

  for oo=1:2:2*NN;

  P(:,oo)=(xxx((oo+1)/2));

  end for ooo=2:2:2*NN;

  P(:,ooo)=(xxx(ooo/2));

  end for oooo=1:4:2*NN;

  Pi(:,oooo)=10;

  end for ooooo=2:4:2*NN;

  Pi(:,ooooo)=-10;

  end for oooooo=3:4:2*NN;

  Pi(:,oooooo)=-10;

  end for ooooooo=4:4:2*NN;

  Pi(:,ooooooo)=10;

  end

  format long e Transposex=xxx'; TranposePsi2=ww'; plot(P,Pi, 'y' ,x1,real(Psi1),xxx,real(Psi2), 'm' ,x3,real(Psi3)); axis([-L 2*L -8 8])

  end case

  3 bar(1:.5:10);

  case

  4 plot(membrane);

  case

  5 surf(peaks);

  end

  % -------------------------------------------------------------------

  • function FileMenu_Callback(hObject, eventdata, handles)

  % hObject handle to FileMenu (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % -------------------------------------------------------------------

  • function OpenMenuItem_Callback(hObject, eventdata, handles)

  % hObject handle to OpenMenuItem (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA)

  file = uigetfile( '*.fig' );

  if ~isequal(file, 0)

  open(file);

  end % -------------------------------------------------------------------

  • function PrintMenuItem_Callback(hObject, eventdata, handles)

  % hObject handle to PrintMenuItem (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA)

  printdlg(handles.figure1)

  % -------------------------------------------------------------------

  • function CloseMenuItem_Callback(hObject, eventdata, handles)

  % hObject handle to CloseMenuItem (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) selection = questdlg([ 'Close ' get(handles.figure1, 'Name' ) '?' ], ...

  [ 'Close ' get(handles.figure1, 'Name' ) '...' ], ...

  'Yes' , 'No' , 'Yes' ); if strcmp(selection, 'No' ) return ; end

  delete(handles.figure1) L=Str2num(get(handles.edit1, 'String' )); Ni=Str2num(get(handles.edit2, 'String' )); ni=Str2num(get(handles.edit6, 'String' )); m=Str2num(get(handles.edit5, 'String' )); iii=Str2num(get(handles.edit9, 'String' )); N=Ni; h=L/(N+1); hplank=6.628*10^-34; omega=8.45*10^20; Vh=0.5*m*omega^2*h^2; Eni=5.6*10^-13*(ni+0.5); k2=(2*m*(Vh-Eni)/hplank^2)^0.5; k1=(2*m*(Eni)/hplank^2)^0.5; Ai=2; x1=linspace(-L,0,200);

  xxx=linspace(0,L,(N+2)); x3=linspace(L,2*L,200); R=(sin(k2*L))/(sin(k2*L)^2+(4*k1^2*k2^2/(k1^2-k2^2)^2)); Bi=(R*Ai^2)^0.5; n=N; Psi1=Ai*exp(i*k1*x1)+Bi*exp(-i*k1*x1); alpha=Ai*exp(i*k1*0)+Bi*exp(-i*k1*0);

  A(ii,n+1)=b(ii,1);

  S=S+A(ii,j)*x(j,1);

  for j=n:-1:ii+1

  S=0;

  for ii=n-1:-1:1

  x(n,1)=A(n,n+1)/A(n,n);

  end end end %Proses Substitusi mundur

  A(ii,k)=A(ii,k)-(m*A(j,k));

  for k=1:(n+1)

  m=A(ii,j)/A(j,j);

  for ii=jj:n

  jj=j+1;

  end end %akhir proses pivot

  u=A(j,p); v=A(j+1,p); A(j+1,p)=u; A(j,p)=v;

  end for j=1:(n-1) % mulai proses pivot if (A(j,j)==0) for p=1:n+1

  %Kalibrasi b kedalam A for ii=1:n

  if Eni<Vh

  A(ii,ii+1)=-1;

  QQi=Ai+Bi; Psi2i=QQi*exp(-k2*xxx); beta=QQi*exp(-k2*L); Fi=beta/(exp(i*k1*L)); Psi3=Fi*exp(i*k1*x3);

  % Psi Analatik

  n=N;

  %Membuat Matriks A Berdiagonal Sesuai Penyelesaian Finit Beda Hingga for ii=1:n

  A(ii,ii)=(2+h^2*((k2)^2));

  end for ii=1:n-1

  end for ii=2:n

  b(n,1)=beta;

  A(ii,ii-1)=-1;

  end % Penyusunan Vektor berdiri b

  b(1,1)=alpha;

  for ii=2:n-1

  b(ii,1)=0;

  end

  end x(ii,1)=(A(ii,n+1)-S)/A(ii,ii);

  end %Menampilkan Vektor ww

  ww=zeros(1,n+2); ww(1,1)=real(alpha); ww(1,N+2)=real(beta);

  for uu=2:N+1

  ww(:,uu)=real(x(uu-1,:))';

  end

  iiii=iii+1; tras=real(Psi2i); format long e Tr=ww; Tra=xxx; AAA=tras'; RT=[Tr' Tra']; tru=Tr(:,iiii); tra=Tra(:,iiii); trasi=tras(:,iiii); warior=abs(tru-trasi); set(handles.edit15, 'String' ,warior); set(handles.edit13, 'String' ,trasi); set(handles.edit14, 'String' ,tra); set(handles.edit10, 'String' ,tra); set(handles.edit11, 'String' ,tru);

  else

  C=(alpha/2)+((i*k1/(2*k2))*(Ai-Bi)); D=(alpha/2)-((i*k1/(2*k2))*(Ai-Bi)); Psi2=C*exp(k2*xxx)+D*exp(-k2*xxx); beta=C*exp(k2*L)+D*exp(-k2*L); Fi=beta/(exp(i*k1*L)); Psi3=Fi*exp(i*k1*x3);

  % Psi Analatik

  n=N;

  %Membuat Matriks A Berdiagonal Sesuai Penyelesaian Finit Beda Hingga for ii=1:n

  A(ii,ii)=(2+h^2*((k2)^2));

  end for ii=1:n-1

  A(ii,ii+1)=-1;

  end for ii=2:n

  A(ii,ii-1)=-1;

  end % Penyusunan Vektor berdiri b

  b(1,1)=alpha;

  for ii=2:n-1

  b(ii,1)=0;

  end

  b(n,1)=beta;

  %Kalibrasi b kedalam A for ii=1:n

  A(ii,n+1)=b(ii,1);

  end for j=1:(n-1) % mulai proses pivot if (A(j,j)==0) for p=1:n+1

  u=A(j,p); v=A(j+1,p); A(j+1,p)=u;

  A(j,p)=v;

  end end %akhir proses pivot

  jj=j+1;

  for ii=jj:n

  m=A(ii,j)/A(j,j);

  for k=1:(n+1)

  A(ii,k)=A(ii,k)-(m*A(j,k));

  end end end %Proses Substitusi mundur

  x(n,1)=A(n,n+1)/A(n,n);

  for ii=n-1:-1:1

  S=0;

  for j=n:-1:ii+1

  S=S+A(ii,j)*x(j,1);

  end

  x(ii,1)=(A(ii,n+1)-S)/A(ii,ii);

  end %Menampilkan Vektor ww

  ww=zeros(1,n+2); ww(1,1)=real(alpha); ww(1,N+2)=real(beta);

  for uu=2:N+1

  ww(:,uu)=real(x(uu-1,:))';

  end

  format long e iiii=iii+1; tras=real(Psi2); Tr=ww; Tra=xxx; AAA=tras'; RT=[Tr' Tra']; tru=Tr(:,iiii); tra=Tra(:,iiii); trasi=tras(:,iiii); warior=abs(tru-trasi); set(handles.edit15, 'String' ,warior); set(handles.edit13, 'String' ,trasi); set(handles.edit14, 'String' ,tra); set(handles.edit10, 'String' ,tra); set(handles.edit11, 'String' ,tru);

  end

Lampiran 2. Daftar Konstanta dan Penyelesaian Analitik

  • 31

  m = 9.11 x 10

  kg = 6.628 x 10

  • 34

  J.s = 8.45 x 10 20 rad/s.

Untuk E < V 1

  Menentukan nilai potensial (V) dengan ( )( ) ( ) 2. Menentukan nilai Energi (E) ( ) dengan ( )( )

  3. Menentukan nilai √ √

  ( )( ) ( )

  4. Menentukan nilai √ ( )

  √ ( )( ) ( )

  Konstanta A = 2 b.

  Kontanta B ( ) dengan i merupakan bilangan imaginer ( √ )

  ( ( ) ( )

  ) c. Kontanta C

  Untuk E &gt; V 1.

  Menentukan nilai potensial (V)

dengan m

( )( ) ( ) Joule

  2. Menentukan nilai Energi (E) ( ) dengan ( )( )

  Joule 3. Menentukan nilai √

  √ ( )( ) ( )

  4. Menentukan nilai √ ( )

  √ ( )( ) ( ) a.

  Konstanta F b.

  Kontanta C ( ) dengan m ( )

  ( ) ( ( ) ) ( ) c.

  Kontanta D ( )

  ( ) ( ) ( ( ) )

  ( ) d. Kontanta A , ( ) ( )-

  , ( ) (

)

  ( )-

  ) ( e.

  Kontanta B , ( ) ( )- , ( ) (

  

)

(

  )- )

  (

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