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Advanced Quantum Mechanics Homeworks Solutions

Thank you Prof. Dr. Cemal Yalabık

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Simple Euler Method

##Usage:Call Octave from terminal ##and then call EulerMethodUmitAlkus.m ##from octave and finally ##press enter. That's all. ##Simple Euler Method ##Constants and initializations x=[]; ## initial empty vector for x y=[]; ## initial empty vector for y x(1)=1; ## initial value of x y(1)=1; ## initial value of y h=1E-3; ## increment in x dery=[]; ## 1st derivative of y wrt x n=1; ## inital loop index for while ## enter the while loop for the interval x=[1,2] while (x(n)<=2) x(n+1)=x(n)+h; dery(n+1)=x(n)*x(n)-2*y(n)/x(n); ##given y(n+1)=y(n)+h*dery(n); ##Euler method n++; endwhile ##exit from the 1st while loop ##Modified Euler Method ##Constant and initializations ymid=[]; ## empty vector function evaluated at x midpoint xmid=[]; ## empty vector func. of midpoints in x ymid(1)=1; ## inital value for ymid. derymid=[]; ## derivative of y at midpoints ##Enter the 2nd while loop n=1; while (x(n)<=2) xmid(n)=x(n)+h/2; derymid(n)=xmid(n)*xmid(n)-2*ymid(...
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FACTORIAL

## Function that calculates the factorial of a number ## Usage : f=factorial(n) function f=factorial(n) ## Initialize the output f=1; ## Check whether the input is correct if ( (n<0) || (rem(n,1)~=0) ) printf("n cannot be a negative number. Exiting...\n"); return endif for num=1:n f*=num; endfor endfunction
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NEWTON’S METHOD FOR MINIMUM

##Newton's Method to find ##the minimum of the function F(x)=(x-2)^4-9 ##with the initial guess xmin=1.0 ##Constants and initializations xmin=[]; ##The empty array of x that minimizes the F(x) xmin(1)=1.0; ##Initial value of the xmin Fmin=[]; ##Minimum values of F(x) x=0.0:0.1:4.0; ##Only for plotting purposes F=[]; ##Our examined Function evaluated on x-space Fp=[]; ##First derivative of F(x) wrt x Fpp=[]; ##Second derivative o F(x) wrt x NSteps=50; ##Step number of iteration ##Algorithm for n=1:NSteps Fmin(n)=(xmin(n)-2)^4-9; Fp(n)=4*(xmin(n)-2)^3; Fpp(n)=12*(xmin(n)-2)^2; xmin(n+1)=xmin(n)-Fp(n)/Fpp(n); Fmin(n+1)=(xmin(n+1)-2)^4-9; endfor printf("x*, at which F(x) is minimum, is %1.6f\n",xmin(n+1)) printf("Minimum of F(x) is %1.6f\n",Fmin(n+1)) F=(x-2).^4-9; subplot(2,1,1) plot(x,F) title('Newton^,s Method-F(x) vs x'); xlabel('x'); ylabel('F(x)'); text(2,-7,'\downarrow') text(1.7,-5.6,'(xmin,Fm...
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