Effect of Various Factors on the Accuracy of Determining the Planck Constant in a Student Physics Laboratory

Abstract

The Planck constant is a fundamental parameter of nature that appears in the description of phenomena on a microscopic scale. Its origin is associated with an explanation of the distribution of the blackbody spectrum performed by Max Planck. This constant stands the basis for the definition of the International System of Units (SI), and, in particular, the new mass definition. This paper presents different methods for determining the Planck constant based on phenomena such as blackbody radiation, light diffraction through a single slit, the current–voltage characteristics of a light-emitting diode, the photoelectric phenomenon, and the hydrogen atom spectrum in the visible range. The Planck constant was measured using instruments in a stationary laboratory and via remote access. The influence of various factors on the accuracy of the measurements was determined, and the consistency of the obtained results with the accepted value of the Planck constant are examined and discussed.

1. Introduction

Determining a correct and reliable value of the Planck constant is an essential point in the study of phenomena at the microscopic level, as well as in the general problem related to the physical constants resulting from the study of the universe. The Planck constant first appeared in the theoretical description of blackbody radiation performed by Max Planck in 1900 [1] (see [2] for historical review). It has been understood that the quantum approach to explaining phenomena at the microscopic level serves the basis of the scientific revolution in the 20th century. The Planck constant, as well as Avogadro’s number, stands the basis for the definition of the SI [3,4]. The Planck constant, h = 6.62607554 × 10⁻³⁴ J·s, can only be determined through physical measurements and cannot be calculated [5]. On the other hand, it has been shown that the value of the Planck constant can be theoretically estimated using the theory of Brownian motion in stochastic mechanics [6].

A common student laboratory method for finding the Planck constant is to study the external photoelectric effect. The theoretical basis of this phenomenon was developed by Albert Einstein in 1905 [7] and experimentally confirmed by Robert Millikan in 1916 [8]. This measurement involves illuminating the metal surface with light of selected wavelengths and determining the corresponding stopping voltages. Next, from the dependence of these voltages on frequency, the Planck constant is determined by fitting the measurement points using, for example, the least-squares method [9,10].

A group of other methods involves the study of blackbody radiation in accordance with the Stefan–Boltzmann law, which allows for the determination of the Planck constant from the Planck radiation law [11]. The role of a blackbody (or rather a gray body) is played by the incandescent filament of a light bulb [12,13,14,15,16,17,18]. In the most simple case, the current–voltage (I–V) characteristic of a light bulb is determined by measuring the voltage and current in a circuit containing an AC (alternative current) power supply. One of the key elements of the system is the light sensor, for example, a suitable phototransistor and a colored light filter (for example, green cellophane) that allows for the selection of a specific wavelength of light. In this experiment, it was also necessary to determine the area of the bulb filament, which is subject to a substantial measurement uncertainty [12]. A similar arrangement was used in a previous paper [13], where several improvements were made in the form of using a low-voltage bulb and a DC (direct current) power supply, selecting an appropriate light filter, and using a gas-filled external photocell. This enabled increased measurement accuracy. Moreover, a comparable measurement method and arrangement were used in Ref. [14].

Another approach to finding the Planck constant is to first determine the Stefan–Boltzmann constant from the I–V characteristics of an incandescent lamp and then calculating the Planck constant from known relationships between the other physical parameters [15,16,18]. The measurement consists of determining the I–V characteristic for the tested bulb, examining the linear dependence of the power dissipated in the bulb’s filament on the fourth power of temperature, and determining the dependence of the power dissipated per square meter of the bulb’s filament on its temperature. Using the least-squares method, the Stefan–Boltzmann coefficient is determined for the linear parts of the measured dependences. Then, using the theoretical description of the phenomenon, the Planck constant can be calculated. The surface area of the bulb filament can be determined using a digital camera [15] or by measuring the resistance of the bulb filament, which allows for the determination of the radius of the tungsten wire and the calculation of the surface area [16]. Moreover, the Planck constant can also be determined by studying the radiation emitted by a Ferry-like blackbody, configured as a double-walled hemisphere [19].

An alternative method often used in student laboratories to determine the Planck constant is the study of the I–V characteristics of light-emitting diodes (LEDs) [20,21,22,23]. This fairly simple method yields the value of the Planck constant that are in general consistent with the standard value. The methods of measuring the Planck constant using LEDs are reviewed, for example, in Ref. [24]. Attention is drawn to significant issues that can affect the accuracy of determining this constant. These issues are related, firstly, to the accuracy of finding the wavelength of radiation emitted by the diodes and, secondly, to the precise measurements of the threshold voltage. It should be noted that diodes do not emit perfectly monochromatic radiation and exhibit a maximum for a specific wavelength. On the other hand, the threshold voltage can be found, for example, by measuring the voltage as soon as the current starts flowing, when the diode starts emitting light, or by determining the threshold voltage from the tangent to the linear part of the I–V characteristic and its intersection with the voltage axis. Also to stress is the process called down-conversion, which is used in the production of LEDs and may contribute to the limitations of this type of diode in measuring the proper value of the Planck constant [25]. These dilemmas can be resolved by conducting precise measurements and comparing the obtained values with the recognized ones.

Moreover, the history and development of methods for determining the Planck constant are discussed in Ref. [26], where the main focus is on using the watt balance technique (WBT). This is one of the most accurate methods that allows for the determination of the precise value of the Planck constant. WBT is a combination of mechanical and electronic measurements that allows for the direct determination of the Planck constant without the need of knowing other fundamental constants [27,28]. Another method for determining the Planck constant uses photoemission spectroscopy [29]. With Einstein’s equation, the Planck constant can be found by precisely measuring the position of the Fermi energy level of a gold atom using light sources with different photon wavelengths. Additionally, a crucial question addresses the physical nature of the Planck constant, which can, for example, be related to the Compton frequency [30].

This paper reviews selected measurement methods based on determining the I–V characteristics of LEDs, the photoelectric effect, the study of the Stefan–Boltzmann law, the validity of Heisenberg’s uncertainty principle, and examining the spectrum of the hydrogen atom. The experimental results presented in this paper were obtained in the Physics Laboratory of Stanisław Staszic State University of Applied Sciences in Piła, Poland (SUAS); studies were conducted using remote experiments [31]. It should be noted that the remote experiments can be performed by any person with basic physics knowledge and has access to a computer or smartphone [32,33]. A comparison of the accuracy of the methods discussed in the context of determining the Planck constant was also made.

2. Photoelectric Effect

In 1905, Einstein has explained the external photoelectric phenomenon, taking into account the corpuscular nature of light and the quantization of energy carried by a photon in accordance with Planck’s postulates [6] (for discussion, see [34,35,36]). The energy of a radiation quantum is equal to

$$E=hf,$$

(1)

where f is frequency. The photons throw electrons out of the metal and give them kinetic energy of E_k. The part of the photon energy is used for the so-called work function W₀, which is related to the binding energy of the electron in the material. Based on the principle of energy conservation, the following relationship can be obtained:

$$hf=E_k+W_0={\displaystyle \frac{m{v}^2}{2}}+W_0 ,$$

(2)

where m = 9.10938291 × 10⁻³¹ kg represents the mass of the electron, and v denotes its velocity. The expression (2) is called the Einstein–Millikan equation [7]. The threshold frequency value, f_p, at which the electron is ejected from the surface of the metal, is

$$h=W_0/f_p .$$

(3)

For most metals, f_p is within the UV (ultra-violet) frequency range. The photoelectric phenomenon occurs for visible light only in alkaline metals. By applying a stopping voltage, V_h, between the anode and photocathode, the maximum speed, v_max, of photoelectrons can be determined from the following equation:

$$v_{\max }=\sqrt{{\displaystyle \frac{2e{V}_h}{m}}} ,$$

(4)

where e = 1.6 × 10⁻¹⁹ C represents an electron charge.

Combining Equations (2) and (4), one finds

$$V_h={\displaystyle \frac{h}{e}}f-{\displaystyle \frac{W_0}{e}} .$$

(5)

Note that the dependence of V_h on f is linear. The value of h/e and W₀/e is by fitting Equation (5) to the measurements. Figure 1 shows a schema of the measurement system, enabling to determine the dependence of the stopping voltage on the frequency of photons incident on the photocathode.

Determining the Current–Voltage Characteristic of the Photocell

In order to study the external photoelectric effect, access to a remote experiment was used [31]. Figure 2 shows the graphical interface of the experiment available on the website [31]. Subsequent changes in the I–V characteristics, depending on the selected wavelength of light, are observed in the measurement window (Figure 2a).

The measurement results are saved in Excel format and displayed on the interface screen. Users can then copy the data to a local computer for further processing (Figure 2b). A description of the method for using the measurement interface is available on the experiment website [31]. A mercury lamp and a set of filters enabling the selection of wavelengths are used to illuminate the photocell, which enables the I–V characteristics of the photocell for a given light frequency to be determined. This experiment uses a PHYWE Systeme GmbH und Co. KG (Göttingen, Germany) photocell with an Sb–Cs (antimony–cesium) cathode that has a spectral response from UV to visible light [37]. An example of an I–V characteristic is shown in Figure 3a. The V_h is determined from the I–V characteristic by finding the maximum voltage for the zero photocurrent for a given wavelength. The measurement results are presented in

Table 1. Data obtained in the photocell experiment. See text for details.

λ (nm)	f (Hz), ×1014	Vh (V)
365	8.21	1.41
405	7.40	1.17
436	6.88	0.86
546	5.49	0.43
568	5.28	0.31

.

Based on the obtained I–V characteristics, the V_h value is determined for each wavelength, λ. The key stage of the measurement is to produce the V_h(f) dependence (Figure 3b). A linear function V_h = 3.74 × 10⁻¹⁵ × f−1.65 fits the obtained measurements using the least-squares method. Based on the obtained values of the fitting parameters from Equation (5), one determines the Planck constant, h*, the work function, W₀, and the threshold frequency, f_p. Therefore, h* = (5.98 ± 0.32) × 10⁻³⁴ J·s, W₀ = (2.64 ± 0.21) ×10⁻¹⁹ J, f_p = W₀/h* = (4.42 ± 0.43) × 10¹⁴ Hz. Using the measurement consistency condition |h*–h| < 2·u(h*), where u(h*) is the measurement uncertainty, gives 0.65 > 0.64. The consistency is not met, but the difference is quite small, and the determined value of the Planck constant can be acceptable in this type of measurement. In conclusion, it should be stated that the main parameter that influences the accuracy of determining the Planck constant using this method is the precision of determining the stopping voltage from the I–V characteristics.

3. Blackbody Radiation

According to the Stefan–Boltzmann law, energy emitted by the blackbody on a unit surface and in unit time is proportional to the fourth power of the absolute temperature of this body [15]. This law can also be applied to the so-called gray bodies whose surfaces show the absorption coefficient, independent of the wavelength smaller than unity. Precise measurements show that often the tungsten filaments of light bulbs emit radiation as a function of temperature according to a power law with an exponent close to 4, as described by the Stefan–Boltzmann law [17,18,38,39]. The formula giving the intensity distribution in the spectrum of radiation of a blackbody was discovered by Planck in the form

$$I(\lambda ,T)={\displaystyle \frac{2c^{2}h}{{\lambda }^5}}{\displaystyle \frac{1}{{\mathrm{e}}^{\frac{hc}{\lambda k_{B}T}}-1}} ,$$

(6)

where c = 299792458 m/s represents speed of light in vacuum, T is the absolute temperature, and k_B = 1.381 × 10⁻²³ J/K is the Boltzmann constant. Integrating Equation (6) within the entire λ wavelength range from 0 to ∞ gives the density of energy flow, I(T):

$$I(T)={\displaystyle {\int }_0^{\infty }I(\lambda ,T)\mathrm{d}\lambda =2\mathrm{\pi }h{c}^2}{\displaystyle {\int }_0^{\infty }\frac{{\lambda }^{-5}}{{\mathrm{e}}^{\frac{hc}{\lambda k_{B}T}}-1}}\mathrm{d}\lambda .$$

(7)

The total emissivity of a blackbody found in this way from Equation (7) has the form of the Stefan–Boltzmann law:

$$I(T)={\displaystyle \frac{2{\pi }^5{k}_B^4}{15c^2{h}^3}}{T}^4=\sigma T^4 ,$$

(8)

where $\sigma ={\displaystyle \frac{2{\pi }^5{k}_B^4}{15c^2{h}^3}}$ = (5.67051 ± 0.00034) × 10⁻⁸ Wm⁻²K⁻⁴ is the constant nowadays named the Stefan–Boltzmann constant. Knowledge of the Stefan–Boltzmann constant allows one to determine the Planck constant from Equation (8), namely:

$$h=\sqrt[3]{{\displaystyle \frac{2{\pi }^5{k}_B^4}{15\sigma c^2}}} .$$

(9)

The proportionality of I~T⁴ is also valid for gray bodies, whose surface has an absorption coefficient independent of the wavelength smaller than 1. Radiation of any body with the absolute temperature T can be described by the non-ideal blackbody dependence:

$$I(T)=\epsilon (\lambda ,T)\sigma S{T}^4 ,$$

(10)

where ε(λ, T) denotes the emissivity of the emitting object and S is an object’s surface (for example, bulb filament). By definition, an ideal blackbody has ε(λ, T) = 1. For instance, a characteristic emissivity of tungsten is about 0.43 [40].

Determination of Current–Voltage Characteristics of Glowing Bulb Filament

The measuring setup (Figure 4a) enables one to determine the resistance, R(t_r), of the bulb’s tungsten filament at room temperature, t_r, and to measure the temperature dependence of the emissivity of the bulb filament heated to high temperatures [41]. In the first step, the resistivity, R(t_r), of the filament is measured at room temperature for quite low current and voltage. This enables the precise determination of the bulb’s resistance at room temperature.

For the tungsten fiber, the following dependence on temperature can be written:

$$R(t)=R_0(1+\alpha t+\beta t^2) ,$$

(11)

where R₀ is the resistance for t = 0 °C and the values of the parameters are: α = 4.82 × 10⁻³ K⁻¹ and β = 6.76 × 10⁻⁷ K⁻² [41]. Then, the filament resistance, R(t_r), determined at room temperature, gives R₀ at 0 °C by solving Equation (11):

$$R_0={\displaystyle \frac{R(t_{\mathrm{r}})}{(1+\alpha t_{\mathrm{r}}+\beta t_{\mathrm{r}}^2)}} .$$

(12)

The temperature of the filament when heated by flowing current is calculated by solving Equation (11) for t; using the relation T = t + 273 gives:

$$T=273+{\displaystyle \frac{1}{2\beta }}\left[\sqrt{{\alpha }^2+4\beta \left({\displaystyle \frac{R(t)}{R_0}}-1\right)}-\alpha \right] .$$

(13)

R(t_r) and R(t) are found by applying Ohm’s law, i.e., by measuring the current and voltage across the filament of the bulb. Equation (13) allows for the conversion of the resistance R(t) to the absolute temperature in Kelvin (see fifth column in

Table 2. Experimental data for the Stefan–Boltzmann law. See text for details.

V (V)	I (A)	P (W)	Vth (mV)	T (K)	logVth	logT
0.44	1.36	0.60	0.04	384.83	−1.36	2.59
1.07	1.91	2.04	0.28	606.26	−0.55	2.78
1.85	2.42	4.48	0.92	787.82	−0.04	2.90
2.08	2.56	5.32	1.19	829.38	0.08	2.92
3.05	3.11	9.49	2.55	971.72	0.41	2.99
4.03	3.61	14.55	4.36	1083.17	0.64	3.03
5.01	4.05	20.29	6.42	1180.04	0.81	3.07
5.99	4.49	26.90	8.80	1256.44	0.94	3.10
7.04	4.90	34.50	11.61	1335.90	1.06	3.13

).

One of the elements (black tube) of the system shown in Figure 4a is a Moll-type thermopile that measures the power emitted by the bulb filament. The sensitivity of the thermopile is constant in the wavelength range from 150 nm to 15 μm (PHYWE 08479.00) and is of about 0.16 mV/mW and the circular detection area is 10 mm. The power is proportional to the voltage V_th on the thermopile and satisfies, within a suitable approximation, the following relationship:

$$V_{\mathrm{th}}=\epsilon \sigma S{T}^4 .$$

(14)

By plotting the dependence of V_th on temperature T in the double-logarithmic scale, one tests the validity of the Stefan–Boltzmann law of energy propagation to the fourth power of temperature:

$$\log V_{\mathrm{th}}=4\log T+\mathrm{const} .$$

(15)

Based on the data from Table 2, the dependence (15) is shown in Figure 4b. Using the least-squares method, a straight line logV_th = 4.53·logT–13.13 is fitted to the measurements. Based on the experimental results, a slope value of n* = 4.53 ± 0.07 is obtained. Using the measurement consistency condition |n* − 4| < 2·u(n*) gives 0.53 > 0.14. Some discrepancy indicates that radiation is probably also lost through thermal conduction. The supplier of the device shown in Figure 4a provides measurement data; based on these data, the exponent for the Stefan–Boltzmann law, equal to 4.19 ± 0.27, is obtained [41]. In this experiment, it is highly critical to accurately determine the initial temperature of the fiber, which was assumed to be the room temperature. In addition, one has to ensure the correctness of electrical contacts and to minimize the resistance of the connecting cables.

There is also another possible method, which involves determining the Stefan–Boltzmann constant and the Planck constant directly from the I–V characteristic for the incandescent bulb. Figure 5a shows such an I–V DC (direct current) characteristic for a tungsten filament bulb (Osram 8100). As can be seen, Ohm’s law is fulfilled for small voltages, less than 0.5 V, but, as the voltage increases, the filament of the bulb starts heating up and the I–V relationship becomes nonlinear. Figure 5b shows the R/R(t_r) dependence of the tungsten filament temperature as calculated from Equation (13). Dependence is linear and characteristic of the tungsten filament [18]. Figure 5c presents the dependence of the power P = IV dissipated by the current flowing through the bulb filament on the absolute temperature to the power of 4. This relationship is linear, especially for higher temperatures, which confirms that the radiation emission is characteristic of a blackbody, so that the following relations are fulfilled:

$$P=\epsilon S\sigma T^4 \mathrm{or} \log {\displaystyle \frac{P}{\epsilon S}}=4\log T+\log \sigma .$$

(16)

Finally, Figure 5d shows the dependence of the power per square meter (Equation (16)) on the absolute temperature in double-logarithmic coordinates. The surface, S, of the bulb filament can be calculated from the measurement of the radius, r, of the filament wire [16] or its mass, m, and resistance, R(t_r), as follows:

$$S=2{\mathrm{\pi }}^2{r}^3{\displaystyle \frac{R(t_{\mathrm{r}})}{\rho }} \mathrm{or} S=\sqrt[4]{{\displaystyle \frac{16{\mathrm{\pi }}^{2}R(t_{\mathrm{r}})m^3}{d^3\rho }}} ,$$

(17)

where ρ = 5.28 × 10⁻⁸ Ωm is tungsten resistivity and d = 19.254 × 10³ kg/m³ is tungsten density. This step of the calculation is subject to quite high uncertainty due to the complication in measuring the filament radius and determining the R(t_r). The filament radius was measured after being removed from the broken bulb and is about 0.085 × 10⁻³ m and R(t_r) = 0.240 Ω (t_r = 298.7 ± 0.1 K). After substituting these values into Equation (17), S = 5.8·10⁻⁵ m². The high-temperature part of Figure 5d is linear and has been fitted with a straight line using the least-squares method. The obtained value of the intercept is found to equal to logσ = −7.273 ± 0.128. This allows one to estimate the magnitude of the Stefan–Boltzmann constant and calculate the Planck constant. The σ value obtained in this way is σ = (5.30 ± 1.56) × 10⁻⁸ W/m²K⁴. The obtained slope of the straight line is 4.27 ± 0.04, which in general confirms the Stefan–Boltzmann law and the feature that the energy emitted from the fiber is thanks to radiation (Equation (16)). The attained value of the Stefan–Boltzmann constant is quite close to that in the literature. However, due to many parameters that are burdened with their uncertainties, this value cannot be decisive. Nevertheless, if using the σ value determined in the experiment, from Equation (9), the Planck constant can be calculated, and then takes the value h = (6.76 ± 0.66) × 10⁻³⁴ J·s. From the consistency, it follows that 0.13 < 1.32, which confirms the consistency of the determined value with the recognized one.

4. Electroluminescence Phenomenon

The phenomenon of electroluminescence occurring, for example, in LEDs, can be used to determine the Planck constant by measuring the I–V characteristics of such a diode. A LED represents a semiconductor device that emits radiation in the visible, infrared, and ultraviolet ranges [42,43]. Electroluminescence occurs as a result of the recombination of holes and electrons in the p-n junction region when it is biased in the direction of conduction. The process of electron transition from a higher to a lower energy level is accompanied by the emission of energy in the form of heat (non-radiative recombination) or light (radiative recombination). The recombination process depends on the type of energy gap. For an oblique energy gap, it is non-radiative recombination, for example, in Si or Ge, while radiative recombination is characteristic of semiconductor materials with a direct energy gap, for example, GaAs, InAs, InP, and InSb. The most commonly used semiconductor material for the production of LEDs is gallium arsenide (GaAs), which is characterized by high quantum efficiency [44]. The intensity of the emitted light depends on the intensity of the current supplied to the junction. The length of the emitted light wavelength λ is described using the formular

$$E_a=hc/\lambda ,$$

(18)

for the width of the bandgap.

Determination of Current–Voltage Characteristics of LED

Many school and university laboratories use the study of the I–V characteristics of LEDs to determine the Planck constant [20,22,23,24,45,46,47]. Here, to determine the Planck constant based on the I–V characteristics of LEDs, the remote experiment [31] is used. The measurement methodology and the obtained results are described below and compared with the experiment conducted in the student physics laboratory at SUAS. The graphic interface of the experiment, available via the website [31], is shown in Figure 6a along with an example of I–V characteristics. The measurement results are automatically saved in an Excel file, which can be copied to a local computer and then processed. Figure 6b shows a device for determining the Planck constant from the study of I–V characteristics of LEDs used in the SUAS physics laboratory.

Figure 7a shows an example of the I–V characteristic for a diode emitting light at a wavelength of λ = 525 nm. The energy required for an electron–hole pair to overcome the band gap of a p–n junction is equal to:

$$E_a=hf=e{V}_p=hc/\lambda ,$$

(19)

where f denotes the frequency of the wave of light and V_p is the threshold voltage at which electrons can overcome the bandgap, which leads to emissions of photons of a specific wavelength, expressed in Equation (18). The V_p was determined from the tangent to the linear part of the I–V characteristic and its intersection with the voltage axis. Based on Equation (19), the Planck constant is equal to

$$h=e{V}_p/f .$$

(20)

The frequency characteristic of a given diode is determined by taking into account the value of the speed of light c and wavelength λ in the maximum spectrum for a given diode: f = c/λ. The V_p was determined from the linear regression fitting to the linear part of the I–V characteristics (Figure 7a). The parameters calculated on the basis of the measurements were collected in

Table 3. Experimental data for LED characteristics (remote experiment). See text for details.

λ (nm)	f (Hz), ×1014	Vp (V)	Ea(J), ×10−19	h(J·s), ×10−34
400	7.50	2.92	4.67	6.23
472	6.36	2.65	4.24	6.67
525	5.71	1.83	2.94	5.15
597	5.03	1.75	2.80	5.57
655	4.58	1.54	2.46	5.38

and

Table 4. Experimental data obtained from the Frederiksen apparatus (SUAS experiment). See text for details.

λ [nm)]	f (Hz), ×1014	Vp(V)	Ea(J), ×10−19	h (J·s), ×10−34
405	7.41	2.89	4.63	6.25
466	6.44	2.72	4.35	6.76
595	5.04	1.89	3.02	6.00
640	4.69	1.79	2.86	6.11
940	3.19	1.10	1.76	5.52

.

The results presented in Table 3 and Table 4 were obtained for LEDs from different manufacturers and two different measurement systems. One finds that the diodes used in the remote experiment (Table 3) give worse values of the Planck constant compared to those in Table 4. The obtained values of the Planck constant for individual diodes show some deviation from the recognized value. Perhaps, this is mainly due to the correctness of determining the threshold voltage from I–V characteristics. The findings lead to an average value of h = (5.80 ± 0.28) × 10⁻³⁴ J·s. However, the average value of the Planck constant obtained based on data from Table 4 is of h = (6.13 ± 0.28) × 10⁻³⁴ J·s. It can be seen that the best consistency of the results is obtained for the wavelengths of about 466–472 nm, while it worsens for longer wavelengths. Figure 7b shows the dependence of activation energy, E_a, on frequency. As soon as this dependence is linear, a linear regression method was used to obtain the Planck constant from a slope coefficient of the straight line fitted to the measuring points. Determined in this way, the Planck constant has a value of (8.08 ± 1.42) × 10⁻³⁴ J·s for Table 3 and (7.13 ± 0.55) × 10⁻³⁴ J·s for Table 4 data, which differs quite significantly from the recognized value. However, the determined values of the Planck constant satisfy the consistency condition compared to the established value of this constant. The parameters of the tested diodes generally have comparable values (for a given wavelength). Moreover, the obtained h values differ quite significantly from the recognized value. It cannot be ruled out that imperfections in the production of LEDs and their structural modifications can lead to the unrealistic or, with considerably high uncertainty, the Planck constants [24,25].

5. Light Diffraction on the Single Slit

The quantum mechanical Heisenberg uncertainty principle was discovered in 1927 [1] (for discussion, see [49,50]). It shows that certain pairs of physical quantities (for example, position and momentum or energy and time) cannot be determined simultaneously with any accuracy. The experiment with the diffraction of light on a single slit allows us to test the correctness of Heisenberg’s uncertainty principle by examining the validity of the following relationship [51]:

$${\displaystyle \frac{z}{\lambda }}\sin \left(\mathrm{arctg}{\displaystyle \frac{y_{1c}}{2l}}\right)=1 ,$$

(21)

where z the slit width, λ is the wavelength of the laser light, y_1c is the distance between the centers of first-order dark fringes, and l is the distance between the sensor and the slit (l = 2152 ± 2 mm).

Measurements can be carried out remotely using the equipment as described on the website [31]. The measurement consists of a narrow or wide slit and laser light with a wavelength of 632 ± 10 nm (red laser) and 532 ± 10 nm (green laser). The intensity of the light in the diffraction pattern is analyzed using a photocell, and the results are saved in an Excel file on a local computer. The examples of measurements are shown in Figure 8a.

The slit width is determined from the diffraction condition; here, for dark fringes,

$$z=2n\lambda l/y_{kc} ,$$

(22)

where y_kc is the distance between the centers of the kth-order dark fringes lying on the right and left of the non-diffracted beam [52].

Based on experimental data (Figure 8b) and using Equation (22), a narrow-slit width is z = (0.129 ± 0.002) mm (data for red laser). This value, along with the remaining experimental data, was used to examine the relationship (21), which gives the value 0.999 ± 0.039, which is quite close to 1. This indicates the validity of the Heisenberg uncertainty principle. The obtained result suggests that the Planck constant obtained in this way takes a value close to the accepted one. However, in this experiment setup, it is not possible to directly determine the Planck constant. But this can be possible if there is a known way of determining a laser photon momentum as an input parameter without prior knowledge of the Planck constant. A possible method for achieving this is to measure the power and the number of photons per unit time of the laser beam [53]. Moreover, quite a simple technique of testing the Heisenberg uncertainty principle with the possibility of determining the Planck constant based on light diffraction was presented in Ref. [54].

6. Determining the Planck Constant Based on the Hydrogen Atom Spectrum

The remote experiment [31] allows one to obtain the hydrogen and mercury atom spectra at a visible band. The location of the spectral lines can also be easily determined by having a grating spectrometer and a spectral lamp containing hydrogen gas in a diluted form in the laboratory. Based on the spectrum of the Balmer series, individual wavelengths for this spectral series are determined in the visible band [55]. By following the description of the remote experiment [31], a hydrogen atom spectrum is obtained (Figure 9a). The length of individual spectral lines in the Balmer series describes the following relationship:

$${\displaystyle \frac{1}{\lambda }}={\displaystyle \frac{m_e{e}^4}{8{\epsilon }_0^2{h}^{3}c}}\left({\displaystyle \frac{1}{2^2}}-{\displaystyle \frac{1}{n^2}}\right)=R_H\left({\displaystyle \frac{1}{2^2}}-{\displaystyle \frac{1}{n^2}}\right) ,$$

(23)

where electron mass m_e = 9.10938291 × 10⁻³¹ kg, electrical permittivity of empty space ε₀ = 8.85417 × 10⁻¹² F/m, and Rydberg’s constant for hydrogen atom R_H = 1.09677 × 10⁷ m⁻¹.

There are four spectral lines in the visible band (Figure 9b): red corresponding to the 3→2 transition, blue-green corresponding to the 4→2 transition, blue-violet corresponding to the 5→2 transition, and violet corresponding to the 6→2 transition. Figure 10 shows the dependence of 1/λ on (1/2²–1/n²), where n = 3, 4, 5. The wavelengths of the spectral lines were read from Figure 9b. From the least-squares fitting of a straight line to the points shown in Figure 10, the Rydberg constant equal to R_H* = (1.0495 ± 0.0429) × 10⁷ 1/m was determined. This value is consistent with the recognized one because the compliance condition is met, namely, 0.047 < 0.086. For example, more precise values of the Rydberg constant were obtained using a grating spectrometer coupled to a digital camera [56] or the optical and computer-aided diffraction-grating spectrometer [57]. Then, the Planck constant can be determined as

$$h=\sqrt[3]{{\displaystyle \frac{m_e{e}^4}{8{\epsilon }_0^{2}c{R_H}^{*}}}} .$$

(24)

After substituting the relevant table data and the determined value of R_H*, the calculated Planck constant is equal to h = (6.72 ± 0.09) × 10⁻³⁴ J·s (the consistency condition is met as soon as 0.09 < 0.18). Uncertainty u(h) can be obtained from the relationship u(h) = (h/3)u(R_H*)/R_H*. This value is quite close to that in the literature, and spectroscopic methods may be recognized as one of the best methods for determining the Planck constant.

7. Discussion

As stressed in Introduction, the Planck constant is a crucial parameter, the precise knowledge of which is the basis for defining other quantities, such as those in the SI system of units [3]. Knowledge of the physical meaning of this constant and the physics behind it can significantly contribute to a better understanding of phenomena occurring on both macroscopic and microscopic scales [5]. The selection of methods for determining the Planck constant presented in this paper is based on professionally available instruments and measurement setups. However, it is also possible to use generally available, inexpensive materials, and especially to take advantage of the technical capabilities of, for example, smartphones [58,59,60,61,62]. The task of physics laboratories is to familiarize students of natural sciences or engineering with measurement techniques also used in scientific laboratories and to teach the methodology of conducting experiments, collecting data, and their interpretation.

Experiments can be conducted both in stationary mode and remotely. Each of these methods has its advantages and disadvantages [32,63]. Direct access to the experimental setup gives students tangible familiarization with the measuring devices and problems that occur during measurements. The students can also independently introduce modifications to the measuring system to improve its operation and improve the accuracy of measurements. In experiments at a distance, it is possible to use existing devices controlled on the web site or simulations of given physical phenomena. The first method is more educational and brings better didactic results. Although the students do not have direct physical contact with the measuring device, they still can observe the system through the installed camera and remotely enter and change measurement parameters. The obtained data are copied to a local computer and processed digitally in the same way as for a stationary experiment [31,64,65]. Remote access to the experiment allows familiarization with the latest measurement technologies and gives an idea of how to conduct professional scientific measurements.

The presented results of the Planck constant measurement were obtained both in remote experiments and in a stationary laboratory (

Table 5. The values of the Planck constant, as determined by different methods. See text for details.

Measurement Method	(h ± u(h)) × 10−34 J·s
Photoelectric effect	5.98 ± 0.32
Blackbody radiation	6.76 ± 0.66
LEDs characteristics	5.80 ± 0.28
	6.13 ± 0.28
	8.08 ± 1.42
	7.13 ± 0.55
Hydrogen atom spectrum	6.72 ± 0.09

). As can be seen, the obtained values of the Planck constant show some scatter and differ more or less from the recognized value. This is not unexpected because physical measurements are burdened with uncertainties resulting from the devices used and the measurement method. The precision of the measurement also lies in the hands of the experimenter, who, after many attempts, becomes familiar with the measurement system and can modify it to eliminate factors that may affect the accuracy of the measurement. An invaluable method in developing measurement results is the least-squares method, the use of which is necessary in every case [66]. Leading the appropriate values to a linear relationship and using the discussed method leads to the optimization of the obtained results, and at the same time allows for the estimation of measurement inaccuracies, which is significant in drawing conclusions as to the correctness of the obtained results.

8. Conclusions

The Planck constant measuring methods are possible to implement in almost every student physics laboratory. The obtained values of the Planck constant by various methods are different from the nominal value within the acceptable measuring uncertainty. The best results are obtained by testing the current–voltage characteristics of LEDs and analyzing the spectroscopy of the hydrogen atom.

Each of the methods discussed here for determining the Planck constant in remote and stationary laboratory conditions has its limitations and advantages. In the remote laboratory, there is little possibility of modifying the measurement arrangement and improving the measurement method. The primary advantage of remote physical measurements is the ability to perform them regardless of location; there is no need for the experimenter to be physically present in the laboratory. In the stationary laboratory, the measurement setup can be adjusted and the measurement system can be continually improved. The most common method for determining the Planck constant is based on examining the current–voltage characteristics of light-emitting diodes. This is justified by the simplicity of the measurement system and the high reliability of the obtained values of the Planck constant. Another method, based on examining the current–voltage characteristics of a light bulb, requires paying attention to the correctness of determining the temperature of the tested system. Methods that utilize the Stefan–Boltzmann law often require knowledge of the size of the light bulb filament, being a critical point in this method due to the accuracy of this measurement. Spectroscopic measurements seem to be highly precise because the determined positions of spectral lines are subject to quite small uncertainties, which contribute to obtaining the Planck constant close to the recognized value. In methods where current–voltage characteristics are studied, the main complication is determining the threshold voltages or stopping voltages based on the graphs. This is the main source of the uncertainty. In methods based on quantum effects, additional measurements need to be used to independently study the momentum of the photon as the latter passes through the slit.