8 Partial dierentials I a unction depends on more than one variable, its rate o change with respect to one o the variables can be determined keeping the others ied The dierential is then a partial dierential The partial dierential o,, z) with respect is denoted,,z the subscript, z indicating that the variables and z are kept constant The second partial dierential with respect to is written 2 The subscripts are oten omitted when it is obvious which variables are held ied The partial dierential o /) need not be with respect to ; it can be with respect to one o the other variables, sa, keeping and z constant It is written,z 8 Functions o two variables The ormal deinition o the partial derivative o a unction, ) with respect to with constant is analogous to equation 23 or a unction o one variable +, ), ), 8) where is the limit o a ver small change in as that change tends to zero A similar equation holds or the partial derivative with respect to with constant and the ininitessimal change d in the unction when both and change, b d and d respectivel, is d d + d 82) The last equation can be used to derive an identit which is oten useul when dealing with unctions o two variables I we put d 0 const) the equation reduces and hence d ) d, 83)
For the smooth unctions usuall met in phsics, which have no singularities where the unction is double-valued, ) ) /, and equation 83 can be written in its usual orm ) ) 84) Eample 8 Determine /) and /) or the unction, ) + ) 2 and use equation 84 to deduce /) Veri the answer b epressing as a unction o and and determining /) directl Hence and rom equation 84 + ) 2 2 + 2 + 2 ) ) 2 + 2; 2 + 2 ) 2 + 2), /) Epressing as a unction o, the solution to the quadratic equation 2 + 2 + 2 0 is 2 2 ± 4 2 4 2 ), or rom which ± 4 /) Problem 8 For well-behaved unctions, Show this to be true or the unction + ) 3 2 82 Functions o three variables Space is three-dimensional, and unctions o three variables are more oten encountered in real situations than unctions o two The partial dierentials o a unction,, z) with respect to one o the variables are now evaluated keeping the other two constant and or completeness two subscripts are needed on the derivative, although or irst dierentials it is eas to see that there is no ambiguit 2
The ininitesimal change in when the variables are changed ininitesimall is d d +,z d + z, dz 85) z, Eample 82 and Show that or the unction ln 3 + 2 + z) 3 z 3 z 3 + 2 + z) 32, 2 3 + 2 + z) 2 62, 3 z 3 + 2 + z) 3 22 Similar dierentiation in the order o irst z then then gives the same answer Problem 82 For the unction o Eample 82 prove that 2 ) [ z ) )] z 83 Partial dierential equations Much o theoretical phsics is ormulated in terms o partial dierential equations, that is, equations involving the partial derivatives o unctions o several variables The equations ma be irst- or second-order and ma be linear or non-linear However, those susceptible to solution in closed orm, that is, solved without recourse to numerical solution using computers, are usuall linear and o these those usuall met are second-order An eample involving two variables is the wave equation or a particle o mass m moving under the inluence o a time-dependent orce which can be represented b a potential V,, z, t) The partial dierential equation or the wave unction is h2 2m + 2 2 + 2 + V,, z, t) j h 2 z 2 t, 86) where h is a constant and j 3
I the potential is independent o time, the phsical interpretation o requires that the equation simpliies to h2 2m + 2 2 + 2 + V,, z) E, 87) 2 z 2 where E is a constant I V /2)k 2 + 2 + z 2 ), the dierential equation to be solved or is h2 2m + 2 2 + 2 + 2 z 2 2 k2 + 2 + z 2 ) E 88) A method which ma be successul or the solution o partial dierential equations and which can be used to help solve the above is the separation o variables Assume a orm o the solution and see i, b substitution in the equation, it results in a new equation in which one side contains all the reerences to one o the variables Since both sides are now independent o each other the must both be equal to a constant and we have an equation involving one variable onl That variable is said to be separated Eample 83 In tring to solve equation 88 assume a solution o the orm X)Y )Zz) and separate the variables 2 Y X )Zz) 2 2, and similarl 2 X)Zz) 2 Y 2 z 2 X)Y Z ) 2 z 2 Substitution in equation 88 gives ) h2 Y Z 2 X 2m 2 + XZ 2 Y 2 + XY 2 Z z 2 + 2 k2 + 2 + z 2 )XY Z E XY Z, and dividing both sides b XY Z leads to h2 2m 2 X X 2 + Y 2 Y 2 + Z Z z 2 + 2 k2 + 2 + z 2 ) E All three variables are now separable resulting in equations o the orm with A and B constants X X 2 + A 2 B, 4
Problem 83 A quantum mechanical particle is constrained to move in a square bo o side a Its wave unctions are given b the solutions o equation 87 and their energies are given b dierent possibilities or the the number E The potential V is zero inside the bo but ininite outside so that the particle s wave unction must be zero at the walls o the bo Show that the smallest value o the energ E is 3π 2 h 2 2ma 2 Choose one o the bottom corners o the square to be at z 0 and remember that since the potential is zero in the bo all possible energies E are greater than zero 5