##### 4.1.5.1 So­lu­tion harme-a

Ques­tion:

Just to check that this book is not ly­ing, (you can­not be too care­ful), write down the an­a­lyt­i­cal ex­pres­sion for and us­ing ta­ble 4.1. Next write down and . Ver­ify that the lat­ter two are the func­tions and in a co­or­di­nate sys­tem that is ro­tated 45 de­grees counter-clock­wise around the -​axis com­pared to the orig­i­nal co­or­di­nate sys­tem.

An­swer:

Take the ro­tated co­or­di­nates to be and as shown:

A vec­tor dis­place­ment of mag­ni­tude in the -​di­rec­tion has a com­po­nent along the -​axis of mag­ni­tude , equiv­a­lent to . Sim­i­larly, a vec­tor dis­place­ment of mag­ni­tude in the -​di­rec­tion has a com­po­nent along the -​axis of mag­ni­tude , equiv­a­lent to . So in gen­eral, for any point ,

Sim­i­larly you get

Turn­ing now to the eigen­func­tions, tak­ing the generic ex­pres­sion

and sub­sti­tut­ing 1, 0, you get

Now sub­sti­tute for and from ta­ble 4.1:

where the con­stant is as given in ta­ble 4.1. You can mul­ti­ply out the ex­po­nen­tials:

The same way, you get

So, the com­bi­na­tion is

Now is ac­cord­ing to the Pythagorean the­o­rem the square dis­tance from the ori­gin, which is the same as . And since , the sum in the com­bi­na­tion eigen­func­tion above is . So the com­bi­na­tion eigen­func­tion is

which is ex­actly the same as above, ex­cept in terms of and . So it is in the ro­tated frame.

The other com­bi­na­tion goes the same way.