Which of the following statements is false?

We begin by recalling the precise scientific facts that correspond to each option and then comparing them with the sentences given in the question. The statement that does not agree with accepted theory will be the false one.

Option A speaks about the splitting of spectral lines when an external electric field is applied. The well-known name for this phenomenon is indeed the Stark effect. Hence the sentence in Option A is in complete accordance with established spectroscopy and is therefore true.

Option B concerns the radiation emitted by a black body as its temperature changes. According to Wien’s displacement law we have

$$\lambda_{\text{max}}\,T = b,$$

where $$\lambda_{\text{max}}$$ is the wavelength at which the emission is most intense, $$T$$ is the absolute temperature and $$b$$ is a universal constant. Rearranging gives

$$\lambda_{\text{max}} = \frac{b}{T}.$$

So, as $$T$$ increases, $$\lambda_{\text{max}}$$ decreases; put in words, the peak of the spectrum shifts from longer to shorter wavelengths, i.e. toward the violet end of the spectrum, or equivalently to higher frequencies. The sentence in Option B says that the “frequency of emitted radiation … goes from a higher wavelength to lower wavelength as the temperature increases.” Although the wording mixes “frequency” and “wavelength” in one clause, its physical content is that the radiation peak moves from longer to shorter wavelengths as temperature rises, which matches Wien’s law. Therefore Option B is also true.

Option C asserts that a photon possesses both momentum and wavelength. The de Broglie relation for a photon is

$$p = \frac{h}{\lambda},$$

where $$p$$ is the momentum, $$h$$ is Planck’s constant and $$\lambda$$ is the wavelength. This clearly endows the photon with a definite momentum whenever its wavelength is finite. Hence Option C is true as well.

Option D states that the Rydberg constant has the unit of energy. To check this, recall the Rydberg formula for the wavenumbers of hydrogen spectral lines:

$$\bar{\nu} = \frac{1}{\lambda} = R\!\left(\frac{1}{n_1^{2}} - \frac{1}{n_2^{2}}\right),$$

where $$\bar{\nu}$$ is the wavenumber, $$\lambda$$ is the wavelength, $$n_1$$ and $$n_2$$ are positive integers with $$n_2 > n_1$$, and $$R$$ is the Rydberg constant. In this equation the term inside the parentheses is dimensionless because it involves only the reciprocals of squared integers. Consequently, the dimensions of $$R$$ must be exactly the same as those of $$\bar{\nu}$$, namely the reciprocal of length:

$$[R] = \frac{1}{\text{length}} = \text{m}^{-1}.$$

A quantity that carries the dimension $$\text{m}^{-1}$$ cannot simultaneously possess the dimension of energy $$\text{(kg·m}^2\text{/s}^2)$$. Therefore the assertion in Option D that “Rydberg constant has unit of energy” is wrong.

Since Options A, B and C are all true and Option D is false, the false statement demanded by the question is Option D.

Hence, the correct answer is Option D.