Example: one-dimensional harmonic oscillator Potential V(x)=(1/2)kx2=(1/2)mo2x2=2T'mvx2 Trial wave function for the ground state p(x)=exp(-cx2) So*HOdx=-( h2/8T2m)Sexp(cx2)d?lexp(-cx2 )dx2 dx +212mv2 x2 exp(-2cx2)dx (h2/4m2m)(πc8)12+m2my2(π8c)l2 ∫φ*ψdx=∫exp(2cx)dx=(/2)l2c12 E6=W=(h2872m)c+(r2)my2/cExample: one-dimensional harmonic oscillator Potential: V(x) = (1/2) kx2 = (1/2) m2x 2 = 2 2m 2x 2 Trial wave function for the ground state: f(x) = exp(-cx2 )  f* H f dx = -(h2 /8 2m)  exp(-cx2 ) d2 [exp(-cx2 )]/dx2 dx + 2 2m 2  x 2 exp(-2cx2 ) dx = (h2 /4 2m) (c/8)1/2 +  2m 2 (/8c3 ) 1/2  f*f dx =  exp(-2cx2 ) dx = (/2)1/2 c -1/2 Ef = W = (h2 /8 2m)c + ( 2 /2)m 2 /c