The feature that is of concern deals with the proper derivatives of the prospective eigenfunctions. This feature can be seen in a simple example.

Consider the continuous function f ( x ) , where − 1 < x < 1 , which is defined as f ( x ) = 0 for − 1 ≤ x ≤ 0 , and f ( x ) = x for 0 ≤ x ≤ 1 .1 A simple review of derivatives shows that f ′ ( x ) = lim ε → 0 [ f ( x + ε ) − f ( x − ε ) ] / 2 ε , which, for our example, leads to f ′ ( x ) = 0 for x < 0 and f ′ ( x ) = 1 for x > 0 . In addition, we find that f ′ ( 0 ) = 1 / 2 . This result has led to f ′ ( x ) becoming a discontinuous function. The next step is defining f ″ ( x ) = lim ε → 0 [ f ′ ( x + ε ) − f ′ ( x − ε ) ] / 2 ε . The result is that f ″ ( x ) = 0 for x < 0 and f ″ ( x ) = 0 for x > 0 . In addition, it follows that f ″ ( 0 ) = lim ε → 0 [ f ′ ( ε ) − f ′ ( − ε ) ] / 2 ε = ∞ . If we imagine that 2 ε → d x , then f ″ ( x ) = δ ( x ) , which is Dirac’s delta function, formally chosen so that ∫ − 1 1     δ ( x ) d x = 1 . Summarizing, we have found that f ″ ( x ) = δ ( x ) . This result has been because f ′ ( x ) is a discontinuous function. Let us add that while ∫ − 1 1 | f ″ ( x ) | d x < ∞ , it follows that ∫ − 1 1 | f ″ ( x ) | 2 d x = ∞ , which is the reason to reject f ″ ( x ) in any Hilbert space.

The proposed eigenfunction ground state of the particle in a box starts with

ϕ 1 ( x ) = 0       ( for   x ≤ − b ) = cos ( π x / 2 b )       ( for   − b ≤ x ≤ b ) = 0       ( for   b ≤ x ) . (2)

However, two derivatives of the eigenfunction ground state, i.e., ϕ 1 ( x ) = cos ( π x / 2 b ) , lead to

ϕ ″ 1 ( x ) = 0       ( for   x < − b ) = ( π 2 / 8 b 2 ) δ ( x + b ) = − ( π 2 / 4 b 2 ) cos ( π x / 2 b )       ( for   − b < x < b ) = − ( π 2 / 8 b 2 ) δ ( x − b ) = 0       ( for   b < x ) . (3)

This equation does not match the desired form of − ℏ 2 ϕ ″ 1 ( x ) = E 1 ϕ 1 ( x ) , and moreover, ∫ − ∞ ∞ | ϕ ″ 1 ( x ) | 2 d x = ∞ , which can not be accepted by any Hilbert space.

While we have only considered the standard ground state, every proposed eigenfunction would have a similar story and none of them could be a proper eigenfunction with a finite eigenvalue, e.g., − ℏ 2 ϕ ″ n ( x ) = E n ϕ n ( x ) with | E n | < ∞ . Nor would they belong to the usual Hilbert space due to having δ ( x ± b ) terms in their second derivative. In the author’s opinion, this behavior fails the conventual quantization of a particle in a box, as it was reviewed in Section 1.

In the next section, we will propose a valid quantization of the particle in a box using an alternate quantization procedure.

The sentence, taken from Section 1, “This connection at x ± b ensures that ϕ ( x ) will be continuous. Clearly, the cos and sin functions fit the requirement, like ϕ n ( x ) = cos ( n π x / 2 b ) , for n = 1 , 3 , 5 , ⋯ , and ϕ n ( x ) = sin ( n π x / 2 b ) , for n = 2 , 4 , 6 , ⋯ ,” makes it clear that only half of the sin and cos terms have been accepted. This means that the standard box treatment leads to only half of the set of eigenfunctions that were presumed.

Some classical problems require reduced coordinates, like the half-harmonic oscillator with the classical Hamiltonian H = ( p 2 + q 2 ) / 2 provided that q > 0 . In so doing, P † ≠ P , and we seek substitutes for p and P. We choose the dilation variable d = p q and q as the new variables. To ensure their independence, we eliminate q = 0 , and then discard all q < 0 , keeping all q > 0 .2 The basic affine quantum operators are now D = ( P † Q + Q P ) / 2 and Q > 0 . Coherent states, used to connect classical and quantum realms [2], have established that with special classical variables we find that H ′ ( d , q ) → H ′ ( D , Q ) . This implies that the classical Hamiltonian can now be H ′ = ( d 2 / q 2 + q 2 ) / 2 , along with q > 0 , and the affine quantum Hamiltonian for this model then becomes

H ′ = ( D Q − 2 D + Q 2 ) / 2 = [ P 2 + ( 3 / 4 ) ℏ 2 / Q 2 + Q 2 ] / 2 . (4)

Happily, the “3/4” term ensures that both P † and ( P † ) 2 act like P and P² in this equation.

This example has proved its validity in several papers that have exposed the eigenfunctions and eigenvalues for the half-harmonic oscillator [3] [4] [5]. This favorable result for the eigenfunctions being of the form ϕ n ( x ) = x 3 / 2 ( p o l y n o m i a l ) n e − x 2 / 2 ℏ , with n = 0 , 1 , 2 , ⋯ , a fact that the first derivative remains a completely continuous function. To see what the x 3 / 2 factor can do for the relevant two derivatives, just examine the simple equation [ − ℏ 2   ( d 2 / d x 2 ) + ( 3 / 4 ) ℏ 2 / x 2 ] x 3 / 2 = ? .

The example of the last section removed q = 0 from the coordinate space. Now we remove two points, namely q = − b and q = b . That leaves three different spaces, and we keep only the middle part, namely, where − b < q < b . We use a suitable dilation variable, namely, d ′ = p ( b 2 − q 2 ) , along with q, which is restricted by − b < q < b . Next, we offer D ′ = [ P † ( b 2 − Q 2 ) + ( b 2 − Q 2 ) P ] / 2 , and − b < Q < b .

Using the new affine variables, the classical Hamiltonian is p 2 = d ′ ( b 2 − q 2 ) − 2 d ′ , which is promoted to

H ′ = D ′ ( b 2 − Q 2 ) − 2 D ′ = P 2 + ℏ 2 [ 2 Q 2 + b 2 ] / [ ( b 2 − Q 2 ) 2 ] . (5)

This ℏ -factor has been selected from a general study of various dilation operators in [6].

It is noteworthy to examine the last equation when Q accepts a Schrödinger representation, i.e., Q = x , and x is extremely close to either ±b. In that case, [ 2 x 2 + b 2 ] / [ ( b 2 − x 2 ) 2 ] ≃ ( 3 / 4 ) / ( b ± x ) 2 , a relation that closely resembles the very strong properties from the present model with those of the former model. Using this fact leads to the suggestion that part of any eigenfunction of (5) is likely to be ψ ( x ) = ( b 2 − x 2 ) 3 / 2 ( r e m a i n d e r ) .3

Efforts to find eigenvalues and eigenfunctions for (5) are open to help shed further information on this effort to find a proper quantization of the particle in a box.