Quantum Fourier Transform (QFT) of a Single Qubit is Hadamard Transform

Below is the definition of QFT as illustrated in the YouTube lecture by Abraham Asfaw.

The LaTex code for the equation is as follows and also available here.

Latex
| \tilde{x} \rangle \equiv ~ QFT ~ |x \rangle ~ \equiv \frac{1}{\sqrt{N}}\sum_{y=0}^{N-1}{e^{\frac{2\pi ix y}{N}}} ~| y \rangle

For the one qubit case, N = 2¹ = 2:

Latex
| \tilde{x} \rangle \equiv ~ QFT ~ |x \rangle ~ \equiv \frac{1}{\sqrt{}N}\sum_{y=0}^{N-1}{e^{\frac{2\pi ix y}{N}}} ~| y \rangle

Latex
\frac{1}{\sqrt{2}}\sum_{y=0}^{1}{e^{\pi ix y}} ~| y \rangle = \frac{1}{\sqrt{2}}[~e^{i \pi x 0}~ | 0 \rangle ~ + ~ e^{i \pi x 1}~| 1 \rangle] = \frac{1}{\sqrt{2}}[~|0\rangle ~+~e^{i \pi x}~|1 \rangle~]

When x = 0:

Latex
QFT~| 0 \rangle = \frac{1}{\sqrt{2}}[~|0\rangle ~+~e^{i \pi 0}~|1 \rangle~] = \frac{1}{\sqrt{2}}[~| 0 \rangle + |1 \rangle~] = |+\rangle

When x = 1:

Latex
QFT~| 1 \rangle = \frac{1}{\sqrt{2}}[~|0\rangle ~+~e^{i \pi 1}~|1 \rangle~] = \frac{1}{\sqrt{2}}[~| 0 \rangle - |1 \rangle~] = |-\rangle

Hence the QFT of a single qubit is essentially the Hadamard transform.