$Assumptions=$Assumptions && {a>0} 
f[z_]=  z Exp[-a z ]
ke=Integrate[f'[z]^2,{z,0,Infinity}]
n=Integrate[f[z]^2,{z,0,Infinity}]
pe=Integrate[z f[z]^2,{z,0,Infinity}]
e=(ke+pe)/n
Plot[e,{a,0,4}] 
FindMinimum[e,{a,1}]
Out[8]= {2.47645, {a -> 0.90856}}
f1[z_]=f[z]/Sqrt[n] /. Last[%]


f[z_]=  z Exp[-a z^2]
ke=Integrate[f'[z]^2,{z,0,Infinity}]
n=Integrate[f[z]^2,{z,0,Infinity}]
pe=Integrate[z f[z]^2,{z,0,Infinity}]
e=(ke+pe)/n
Plot[e,{a,0,4}] 
FindMinimum[e,{a,.2}]
Out[19]= {2.34478, {a -> 0.260531}}
f2[z_]=f[z]/Sqrt[n] /. Last[%]


f[z_]=  z Exp[-a z^(3/2)]
ke=Integrate[f'[z]^2,{z,0,Infinity}]
n=Integrate[f[z]^2,{z,0,Infinity}]
pe=Integrate[z f[z]^2,{z,0,Infinity}]
e=(ke+pe)/n
Plot[e,{a,0,4}] 
FindMinimum[e,{a,.4}]
Out[27]= {2.34723, {a -> 0.471405}}
f3[z_]=f[z]/Sqrt[n] /. Last[%]

f[z_]=AiryAi[z+AiryAiZero[1]]
n=Integrate[f[z]^2,{z,0,Infinity}]
f4[z_]=f[z]/Sqrt[n] 

N[-AiryAiZero[1]]
Out[33]= 2.33811

Plot[{f1[z],f2[z],f3[z],f4[z]},{z,0,5}]
(blue,orange,green,red)
