Conventionally wavefunctions are displayed,as here, on adiagramshowingthepotential energyfunction,withthe16zero for each wavefunction at the level of its energy.1. Note that the lowest-energy wavefunction has noFh?nodes (points where the wavefunction is zero) except atthe8ma2ends of the box where the zero is required by the boundary3condition.Thenextwavefunctionhas1node,thenexthas2and so on, each wavefunction having one morenode thanthe previous one.2. even function (ground state)41oddfunction(thefirstexcitedstate)a0even (thesecond)odd (the third)Wavefunctions foraparticleina box
Wavefunctions for a particle in a box Conventionally wavefunctions are displayed, as here, on a diagram showing the potential energy function, with the zero for each wavefunction at the level of its energy. 1. Note that the lowest-energy wavefunction has no nodes (points where the wavefunction is zero) except at the ends of the box where the zero is required by the boundary condition. The next wave function has 1 node, the next has 2, and so on, each wavefunction having one more node than the previous one. 2. even function (ground state) odd function (the first excited state) even (the second) odd (the third)
Sets of eigenfunctions3. A general property (proved later) of the set of eigenfunctions of an operator like theHamiltonian is that they are orthogonal; that is,Jymy,dx= 0if m +n.In the present case, the orthogonality is easily demonstrated:(m+n)元x(m-n)元xm元x.n元xXsinndxsincOSCOSaaaaItisnoweasytoshowthattheresultiszerounlessm=n.If the wavefunctions are normalised, so that m*mdx = 1 for all m, thenJymy,dx =Omn (Kronecker delta)and the set is said to be orthonormal
3. A general property (proved later) of the set of eigenfunctions of an operator like the Hamiltonian is that they are orthogonal; that is, In the present case, the orthogonality is easily demonstrated: It is now easy to show that the result is zero unless m = n. If the wavefunctions are normalised, so that 𝜓𝑚 ∗𝜓𝑚d𝑥 = 1 for all m, then (Kronecker delta) and the set is said to be orthonormal. * d 0 m n x if 𝑚 ≠ 𝑛. 0 0 1 sin sin d cos cos d 2 a a m x n x m n x m n x x x a a a a * d m n mn x Sets of eigenfunctions
Expansionin eigenfunctions4.Another important property is that any function of the same variables with the sameboundary conditions can be expressed as a linear combination of the n :y=Ec.VnTo find the coefficients we just multiply the above equation by m* and integrate:'ymydx=Ec.J"ymy.dx=cmfymy.dxsince all other terms in the sum on the right vanish because of the orthogonalityIf the m are normalized this just reducesCm=J,wmvdx
Expansion in eigenfunctions 4. Another important property is that any function of the same variables with the same boundary conditions can be expressed as a linear combination of the 𝝍𝒏 : To find the coefficients we just multiply the above equation by 𝜓𝑚 ∗ and integrate: since all other terms in the sum on the right vanish because of the orthogonality. If the 𝜓𝑚 are normalized this just reduces n n n c * * 0 0 * 0 d d d a a m n m n n a m m m x c x c x * 0 d a m m c x