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Wave function and its first derivative must be continuous becaus wave function is solution of Schroedinger equation:

Let's examine one dimensional case.

## \frac{d^2 \psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x) ##

David J. Griffiths gives a problem in his quantum mechanics book

http://depts.washington.edu/chemcrs/bulkdisk/chem551A_win06/exam_answers_midterm_answer_problem2.pdf [Broken]

Here is very similar problem. That in infinite potential well function ##\psi(x)=Ax## for ##0 \leq x \leq \frac{a}{2}## and ##\psi(x)=A(a-x)## for ##\frac{a}{2} \leq x \leq a##. Otherwise ##\psi(x)=0##. This wave function do not have a derivative in point ## x=\frac{a}{2}##. Is it possible that we choose wave function of this type?

Let's examine one dimensional case.

## \frac{d^2 \psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x) ##

David J. Griffiths gives a problem in his quantum mechanics book

http://depts.washington.edu/chemcrs/bulkdisk/chem551A_win06/exam_answers_midterm_answer_problem2.pdf [Broken]

Here is very similar problem. That in infinite potential well function ##\psi(x)=Ax## for ##0 \leq x \leq \frac{a}{2}## and ##\psi(x)=A(a-x)## for ##\frac{a}{2} \leq x \leq a##. Otherwise ##\psi(x)=0##. This wave function do not have a derivative in point ## x=\frac{a}{2}##. Is it possible that we choose wave function of this type?

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